Modeling the Orimet multiphysical flow of fresh self-compacting concrete considering proportionate heterogeneity of aggregates

Abstract

Filling ability is one of the prominent rheological properties of the self-compacting concrete (SCC), which has been studied in this research work deploying the functional behavior of the concrete through the Orimet apparatus using the coupled ANSYS-SPH interface. Seven (7) model cases: Case 1: 0% coarse particles and 100% fine particles, Case 2: 60% coarse particles and 40% fine particles, Case 3: 55% coarse particles and 45% fine particles, Case 4: 50% coarse particles and 50% fine particles, Case 5: 45% coarse particles and 55% fine particles, Case 6: 40% coarse particles and 60% fine particles and Case 7: 100% coarse particles and 0% fine particles were studied and optimized. The maximum size of the coarse aggregates considered is 20 mm and that of the fine aggregates is below 4 mm. The Bingham model properties for the multiphysics (SPH)-ANSYS models’ simulation are; Viscosity = 20 ≤ μ ≤ 100 and the Yield stress = 50 $\le {\tau }_0\le 200,$ standard flow time, t (s) ranges; 0 ≤ t ≤ 6, and the Orimet volume is 10 l. The minimum boundary flow time, which represents the time (0 ≤ t ≤ 6) it takes for the SCC to completely flow through a specified distance, typically measured in seconds was modeled for in the seven (7) model cases. The first case; 0%C mixed with 100%F flowed out completely in 6 s, second case; 40%C mixed with 60%F completely flowed out in 5 s, third case; 45%C mixed with 55%F completely flowed out in 9 s, fourth case; 50%C mixed with 50%F completely flowed out in 12 s, fifth case; 55%C mixed with 45%F completely flowed out in 11 s, sixth case; 60%C mixed with 40%F completely flowed out in 12 s, and lastly, the 7th case; 100%C mixed with 0%F completely flowed out in 20 s. The minimum flow time was considered alongside other relevant parameters and tests, such as slump flow, passing ability, segregation resistance, and rheological properties (stresses), to comprehensively assess the filling ability of SCC in this model. By considering these factors and the optimized mix (40%C + 60%F:5 s), engineers and researchers can optimize the SCC mix design to achieve the desired flowability and filling performance for their specific construction applications. The multiphase optimized mix (40%C + 60%F:5 s) was further simulated using the coupled interface of the ANSYS-SPH platform operating with the CFX command at air temperature of 25 °C, which incorporated the studied density of 2400 kg/m³, plastic viscosity boundary, yield stress, and aggregate sampling. The model simulation operated on total number of nodes = 143,083, total number of elements = 753,292, total number of tetrahedrons = 753,292, and total number of faces = 68,488, and produced Dynamic Viscosity = 1.831E-05 kg m⁻¹ s⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Absorption Coefficient = 0.01 m⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Refractive Index = 1.0 m m⁻¹, Molar Mass = 1 kg kmol⁻¹, Specific Heat Capacity = 8.80E + 02 J kg⁻¹ K⁻¹, Normal Speed = 165 mm s⁻¹, Pressure Profile Blend = 0.05, and Maximum Partition Smoothing Sweeps = 100. Also, the Global Length = 1.9144E-01, Minimum Extent = 1.1800E-01, Maximum Extent = 6.5976E-01, Density = 1.1850E + 00, Velocity = 1.6500E-01, Advection Time = 1.1602E + 00, and Reynolds Number = 2.0443E + 03. The simulation also produced wall forces and moments on the wall of the Orimet for the optimized mix containing 40%C + 60%F:5 s flow mix as follows; pressure force on wall; −3.0996E-08, −2.0863E-07, and −3.5048E-04 for x-component, y-component, and z-component, respectively, viscous force on wall; −5.5332E-10, −9.2298E-10, and −2.7250E-05 for the x-, y-, and z-components, respectively, pressure moment on wall; −5.0051E-05, 3.1362E-06, and 2.2774E-09 for the x-, y-, and z-components, respectively and viscous moment on wall; −3.8925E-06, 2.4396E-07, and −7.2693E-11 for the x-, y-, and z-components, respectively. Also, the maximum residuals were located at node 110413 for the pressure, node 76766 for the K-TurbKE, and node 110724 for the E-Diss.K. Ideally, the mix, 40%C + 60%F:5 s has been proposed as the mix with the most efficient flow to achieve the filling ability for sustainable structural concrete construction.

Keywords:

• Self-compacting concrete (SCC)
• multiphysical flow (MF)
• Orimet flow time
• aggregate homogeneity
• SPH
• ANSYS
• Non-Newtonian viscous fluid

Reviewing Editor:

• Sanjay Kumar Shukla
• Edith Cowan University
• Australia

Subjects:

• Fluid Mechanics
• Materials Science
• Civil, Environmental and Geotechnical Engineering

1. Introduction

The multiphysical flow of fresh self-compacting concrete (SCC) takes into account the proportionate heterogeneity of fine and coarse aggregates (Onyelowe, Kontoni, Onyia, et al., 2023). SCC is a specialized type of concrete that is highly workable and able to flow and fill complex forms under its own weight without the need for mechanical consolidation (Kaveh et al., 2018; Onyelowe, Kontoni, Ebid, et al., 2023). In SCC, the proportionate heterogeneity of fine and coarse aggregates refers to the variation in particle sizes and distributions within the aggregate mixture (Franz & Wendland, 2018; Kaveh et al., 2018). This heterogeneity can affect the flow behavior and stability of the SCC mix, as well as its mechanical properties. To model the multiphysical flow of fresh SCC considering the proportionate heterogeneity of aggregates, various approaches can be used. One common approach is to employ computational fluid dynamics (CFD) simulations (Colagrossi et al., 2009; Lind et al., 2020; Monaghan, 2005; Petersson et al., 1996; Sathyan et al., 2018). CFD simulations can provide insights into the flow behavior of SCC by solving the governing equations of fluid dynamics, such as the Navier-Stokes equations, within the given geometry (Onyelowe, Kontoni, Onyia, et al., 2023). In the case of SCC, the heterogeneity of aggregates can be accounted for by using a particle-based model. This involves representing each aggregate particle as a discrete entity with its own properties, such as size, shape, and density (Colagrossi et al., 2009; Lind et al., 2020; Monaghan, 2005; Petersson et al., 1996; Sathyan et al., 2018). The interaction between particles and the surrounding fluid (cement paste) can be simulated using appropriate numerical techniques, such as the discrete element method (DEM) or smoothed particle hydrodynamics (SPH).

By incorporating the proportionate heterogeneity of fine and coarse aggregates into the particle-based model, it is possible to investigate how the variation in particle size and distribution affects the flow characteristics of SCC (Onyelowe et al., 2022a, 2022b; Onyelowe, Naghizadeh, et al., 2023; Onyelowe & Kontoni, 2023). This can include studying phenomena such as particle segregation, differential settling, and flowability variations within the mixture. Experimental validation of the simulation results is crucial to ensure the accuracy and reliability of the model. This can involve conducting rheological tests, such as slump flow tests or V-funnel tests, to measure the flowability and stability of the SCC under different conditions (Gram & Silfwerbrand, 2011; Onyelowe et al., 2022a; Onyelowe, Kontoni, Ebid, et al., 2023; Onyelowe, Kontoni, Onyia, et al., 2023; Toosi et al., 2015). The experimental data can then be compared with the simulation results to validate the model and make necessary adjustments if required. Overall, modeling the multiphysical flow of fresh SCC considering the proportionate heterogeneity of fine and coarse aggregates is a complex task that requires a combination of numerical simulations and experimental validation (Onyelowe et al., 2022a). By accurately capturing the flow behavior of SCC, these models can help optimize the mixture design and improve the overall performance of self-compacting concrete in various applications (Petersson et al., 1996). The multiphysical flow time of fresh self-compacting concrete (SCC) can be evaluated using a rheometer (Gram & Silfwerbrand, 2011; Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015). A rheometer is an instrument specifically designed to measure the rheological properties of materials, including flow characteristics. When considering the proportionate heterogeneity of fine and coarse aggregates in SCC, the rheometer can provide valuable information about the flow behavior and workability of the concrete mixture (Kaveh et al., 2018). It allows for the measurement of parameters such as yield stress, viscosity, and shear rate, which are essential in assessing the flow properties of SCC (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015; Wang et al., 2016). To evaluate the multiphysical flow time of fresh SCC using a rheometer, the following steps can be followed: (i) Sample Preparation: Prepare a representative sample of fresh SCC, considering the desired proportionate heterogeneity of fine and coarse aggregates. Ensure that the sample is properly mixed to achieve a homogenous distribution of aggregates. (ii) Test Setup: Set up the rheometer according to the manufacturer's instructions. This includes selecting the appropriate geometry (e.g. concentric cylinders or parallel plates) and adjusting the gap or height between the measuring surfaces. (iii) Sample Loading: Place the SCC sample onto the measuring surface of the rheometer and ensure proper alignment. (iv) Test Procedure: Initiate the test by applying a controlled shear rate or stress to the SCC sample.

The rheometer will measure the response of the material in terms of shear stress and strain rate. (v) Data Collection: The rheometer will provide real-time data on the shear stress and shear rate. These data can be used to analyze the flow behavior and workability of the SCC (Franz & Wendland, 2018; Kaveh et al., 2018). (vi) Analysis: Analyze the obtained data to determine various flow properties of the SCC, such as flow time, viscosity, and yield stress. These properties can provide insights into the material's ability to flow and fill complex forms. By considering the proportionate heterogeneity of fine and coarse aggregates in the SCC mixture, the rheometer can help assess the impact of aggregate distribution on the flow properties and workability of the concrete. This information is crucial for optimizing the mixture design and ensuring the desired performance of SCC in practical applications. Moreover, this suggests the best heterogenous proportions of coarse and fine aggregate with the most efficient flow time and passing ratio for more sustainable concrete handling.

2. Important considerations

2.1. Numerical modelling of multiphysical flow time considering the proportionate heterogeneity of fine and coarse aggregates

Numerical modeling of the multiphysical flow time of fresh self-compacting concrete (SCC) considering the proportionate heterogeneity of fine and coarse aggregates involves simulating the flow behavior of SCC using computational methods (EFNARC, 2002; Onyelowe, Kontoni, Onyia, et al., 2023). This approach allows for a detailed analysis of the complex interactions between the fluid (cement paste) and the solid particles (aggregates) within the mixture (Kaveh et al., 2018). Here are the general steps involved in numerical modeling of the multiphysical flow time of fresh SCC: (i) Geometry and Mesh Generation: Create a three-dimensional computational domain that represents the physical space where the SCC flow occurs (Onyelowe et al., 2022a). This domain should include the boundaries and the desired proportionate heterogeneity of fine and coarse aggregates. Generate a mesh that discretizes the domain into small elements to facilitate numerical calculations. (ii) Governing Equations: Formulate the governing equations that describe the flow of the SCC mixture (Onyelowe, Kontoni, Onyia, et al., 2023). These typically include the continuity equation and the Navier-Stokes equations, which govern fluid flow. Additional equations may be needed to account for particle-particle interactions, particle-fluid interactions, and the effects of aggregate heterogeneity. (iii) Constitutive Models: Define constitutive models that capture the rheological behavior of the SCC mixture, including the flow properties of the cement paste and the interactions between the fluid and solid particles (Gram & Silfwerbrand, 2011; Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015). These models should account for the proportionate heterogeneity of fine and coarse aggregates, such as variations in particle size and distribution. (iv) Boundary Conditions: Specify appropriate boundary conditions that represent the flow conditions at the inlet and outlet of the computational domain (Franz & Wendland, 2018; Kaveh et al., 2018). These conditions may include prescribed velocities, pressures, or flow rates. (v) Numerical Solution: Discretize the governing equations and solve them numerically using suitable numerical methods, such as finite element methods or finite difference methods. This involves iterating through time to simulate the evolving flow behavior of the SCC mixture. (vi) Post-Processing: Analyze the simulation results to extract relevant information about the flow time of the fresh SCC. This can include evaluating flow velocities, pressure distributions, particle trajectories, or other quantities of interest (Gram & Silfwerbrand, 2011; Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015). Compare the numerical results with experimental data or reference models to validate the accuracy of the simulation. It's important to note that the numerical modeling of SCC flow is a complex task that requires expertise in computational fluid dynamics (CFD) and appropriate software tools (Onyelowe, Kontoni, Onyia, et al., 2023). The accuracy of the simulation results relies on the quality of the input data, including material properties, boundary conditions, and the representation of aggregate heterogeneity. Numerical modeling can provide valuable insights into the flow behavior and workability of SCC, allowing for optimization of mixture design and prediction of performance in practical applications (Colagrossi et al., 2009; Onyelowe et al., 2022a). However, it should be complemented with experimental validation to ensure the reliability of the model and its predictions

2.2. Theory of the multiphysical flow of self-compacting concrete

The multiphysical flow theory of fresh self-compacting concrete (SCC) refers to a conceptual framework that considers the simultaneous influence of multiple physical phenomena on the flow behavior of SCC (Onyelowe et al., 2022a; Onyelowe, Kontoni, Ebid, et al., 2023; Onyelowe, Kontoni, Onyia, et al., 2023). It recognizes that the flow of SCC is governed by not just one physical process, but a combination of several interacting mechanisms. In the context of the multiphysical flow theory, these physical phenomena can include: (i) Rheology: Rheology deals with the flow and deformation of materials (Onyelowe, Kontoni, Onyia, et al., 2023). In the case of SCC, it involves the study of viscosity, yield stress, and shear thinning behavior (Onyelowe, Kontoni, Ebid, et al., 2023). The multiphysical flow theory considers how the rheological properties of the concrete mixture influence its flow characteristics. (ii) Surface tension and cohesion: Surface tension refers to the cohesive forces between liquid molecules at the concrete-air interface (Onyelowe et al., 2022a; Onyelowe, Kontoni, Ebid, et al., 2023). Cohesion refers to the internal forces within the concrete material itself (Kaveh et al., 2018). The multiphysical flow theory takes into account how surface tension and cohesion affect the flow of SCC, influencing its ability to fill and maintain shape without segregation. (iii) Particle interaction and packing: The behavior of aggregates and other solid particles within the SCC mixture is essential to its flow properties (Kaveh et al., 2018). The multiphysical flow theory considers how particle interactions, such as friction, collision, and interlocking, affect the flow and packing of SCC. (iv) Interfacial forces: Interfacial forces arise at the interfaces between different materials within the SCC mixture, such as between cement particles and water or between cement paste and aggregates (Colagrossi et al., 2009). These forces play a role in governing the flow behavior and stability of SCC. (v) Fluid dynamics: Fluid dynamics principles, such as conservation of mass and momentum, are applied to analyze the flow of SCC. The multiphysical flow theory incorporates these principles to understand the flow behavior of SCC in various geometries and under different flow conditions. By considering the multiphysical aspects of SCC flow, this theory aims to provide a comprehensive understanding of the complex interactions and mechanisms that govern the behavior of fresh self-compacting concrete (Gram & Silfwerbrand, 2011; Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015). It recognizes that multiple physical phenomena interact simultaneously, influencing the flow properties, filling ability, and stability of SCC mixtures. It's important to note that the multiphysical flow theory is a theoretical framework aimed at enhancing our understanding of SCC behavior. It may involve mathematical modeling, computational simulations, and experimental investigations to explore the intricate nature of SCC flow.

2.3. Theory of multiphysical flow considering proportionate heterogeneity of aggregates

The theory of the multiphysical flow of fresh self-compacting concrete considering proportionate heterogeneity of fine and coarse aggregates focuses on the impact of the distribution and proportion of fine and coarse aggregates on the flow behavior of self-compacting concrete (SCC). It recognizes that the size and distribution of aggregates can significantly influence the flow properties and rheology of SCC (Gram & Silfwerbrand, 2011; Hosseinpoor et al., 2017; Sigalotti et al., 2021; Sonebi, 2004; Sonebi et al., 2007; Sonebi & Bartos, 2002; Toosi et al., 2015). In this theory, the proportionate heterogeneity of fine and coarse aggregates refers to the deliberate variation in the size, shape, and gradation of aggregates within the SCC mixture. By carefully designing the aggregate composition, it is possible to influence the flow behavior of SCC and achieve the desired properties (Franz & Wendland, 2018; Kaveh et al., 2018). The following are key aspects considered in this theory: (i) Particle packing: The proportionate heterogeneity of fine and coarse aggregates affects the packing density and arrangement of particles within the SCC mixture. By optimizing the gradation and distribution of aggregates, it is possible to achieve better particle packing, which can enhance the flowability and filling ability of SCC. (ii) Void content and water demand: The distribution and proportion of fine and coarse aggregates influence the void content within the SCC mixture (Onyelowe, Kontoni, Onyia, et al., 2023). A well-designed aggregate composition can help minimize the voids while maintaining an adequate amount of water for proper hydration. This balance is crucial for achieving the desired flow properties without compromising the strength and durability of the hardened concrete. (iii) Interparticle friction and lubrication: The interaction between fine and coarse aggregates plays a role in the flow behavior of SCC (Franz & Wendland, 2018; Kaveh et al., 2018). The proportionate heterogeneity affects the interparticle friction and lubrication, which can affect the flowability and viscosity of the concrete mixture. By carefully selecting the aggregate sizes and gradation, it is possible to control these interactions and optimize the flow properties. (iv) Segregation resistance: The proportionate heterogeneity of aggregates can influence the resistance to segregation in SCC. By designing the gradation and distribution of aggregates, the theory aims to minimize particle segregation, ensuring uniform distribution of aggregates and homogeneity of the concrete mixture during the flow process (Onyelowe, Kontoni, Onyia, et al., 2023). By considering the proportionate heterogeneity of fine and coarse aggregates, the theory of the multiphysical flow of fresh self-compacting concrete offers a framework for optimizing the aggregate composition to achieve desired flow properties, filling ability, and stability of SCC (Gram & Silfwerbrand, 2011; Onyelowe et al., 2022a; Wang et al., 2016). It recognizes that the distribution and proportion of aggregates have a significant influence on the flow behavior and performance of SCC mixtures. It's worth noting that the theoretical concepts and models associated with this theory may involve computational simulations, experimental investigations, and empirical observations to study the complex interactions between aggregates and the fresh concrete matrix.

2.4. Modeling multiphysical flow of self-compacting concrete using coupled ANSYS and SPH

Modeling the multiphysical flow of self-compacting concrete (SCC) using ANSYS and Smoothed Particle Hydrodynamics (SPH) can be a complex task, but it is certainly achievable (Colagrossi et al., 2009). Here's an outline of the general steps involved in the process: (i) Define the Geometry: Start by creating a 3D geometry of the domain in ANSYS (Kaveh et al., 2018). This includes the boundaries, such as walls and any other structures involved in the flow of SCC. (ii) Mesh Generation: Generate a suitable mesh for the domain. In the case of SCC, you may consider using a tetrahedral or hexahedral mesh, depending on the complexity of the geometry and the level of detail required (Lind et al., 2020). (iii) Material Properties: Define the material properties of SCC in ANSYS. This includes parameters such as density, viscosity, and other rheological properties (Onyelowe, Kontoni, Onyia, et al., 2023). These properties are essential for accurately modeling the flow behavior of SCC. (iv) Fluid Flow Simulation: Set up the fluid flow simulation using the ANSYS Fluent module (Petersson et al., 1996). Specify the boundary conditions, such as inlet and outlet velocities or pressures. You may also need to consider any additional forces or sources acting on the SCC, such as gravity or external pumps. (v) SPH Particle Setup: To incorporate SPH into the simulation, you need to define the particles that represent the SCC material (Colagrossi et al., 2009). ANSYS provides capabilities to create and track SPH particles within the simulation domain (Kaveh et al., 2018). (vi) Coupling ANSYS and SPH: Establish the coupling between ANSYS Fluent and the SPH solver. This involves exchanging information between the fluid flow simulation and the SPH particle representation. The coupling allows the fluid flow simulation to influence the movement and behavior of the SPH particles and vice versa (Colagrossi et al., 2009; Onyelowe, Kontoni, Ebid, et al., 2023). (vii) Simulation and Analysis: Run the simulation and analyze the results. ANSYS provides various post-processing tools to visualize and analyze the flow behavior, such as velocity profiles, pressure distributions, and material distribution within the concrete (Monaghan, 2005). (viii) Validation and Optimization: Validate the simulation results against experimental data or existing analytical solutions, if available. If the results are satisfactory, you can further optimize the simulation setup to improve accuracy or efficiency (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). It's important to note that modeling multiphysical flow of SCC using ANSYS and SPH requires a good understanding of both finite element analysis and SPH methods, as well as expertise in using ANSYS software. It's recommended to consult relevant literature, research papers, or seek guidance from experts in the field to ensure accurate and reliable simulations.

2.5. Constitutive equations of multiphysical flow of self-compacting concrete

The multiphysical flow behavior of self-compacting concrete (SCC) is typically described by a combination of several constitutive equations that consider various physical phenomena occurring in the material (Onyelowe, Kontoni, Onyia, et al., 2023). These equations capture the mechanical, rheological, and thermal aspects of the flow. Here are the key constitutive equations used in modeling the multiphysical flow of SCC: (i) Bingham Plastic Model: The Bingham plastic model, as mentioned earlier, describes the flow behavior of SCC once it surpasses the yield stress. It relates the shear stress (τ) to the shear rate (γ) using the yield stress (τ₀) and plastic viscosity (μ). (ii) Continuity Equation: The continuity equation represents the conservation of mass and is commonly used in fluid dynamics (Onyelowe et al., 2022a, 2022b; Onyelowe, Naghizadeh, et al., 2023; Onyelowe & Kontoni, 2023). It states that the rate of change of mass within a control volume is equal to the net mass flow rate into or out of the control volume. (iii) Navier-Stokes Equations: The Navier-Stokes equations are fundamental equations in fluid dynamics, describing the conservation of momentum. They account for the effects of viscosity, pressure, and body forces on the fluid flow. The equations consist of the conservation of mass and the conservation of momentum equations. (iv) Thermal Energy Equation: The thermal energy equation considers the heat transfer within the SCC material. It accounts for thermal conduction, convection, and heat generation or absorption due to chemical reactions or other sources (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). This equation provides information about the temperature distribution and its impact on the flow behavior. (v) Rheological Models: SCC often exhibits time-dependent flow behavior, and various rheological models can be employed to capture this aspect. Examples include the Herschel-Bulkley model, the Power Law model, and the Cross model. These models introduce additional parameters to describe the time-dependent viscosity or shear thinning/thickening behavior of SCC. (vi) Solid Mechanics Equations: In addition to fluid flow equations, solid mechanics equations may be employed to model the behavior of solid particles or aggregates within SCC. These equations consider the deformation and stress distribution within the solid phase of the material (Gram & Silfwerbrand, 2011; Onyelowe et al., 2022a; Wang et al., 2016). It's important to note that the specific choice and combination of constitutive equations can vary depending on the modeling approach and the level of detail required for a particular analysis. The parameters in these equations, such as yield stress, plastic viscosity, thermal properties, and rheological parameters, are typically obtained through experimental testing and characterization of SCC.

2.6. Mathematics of multiphysical flow of self-compacting concrete

The mathematics of modeling the multiphysical flow of self-compacting concrete (SCC) involves describing the various physical phenomena that occur during the flow process. Here are some key mathematical concepts and equations commonly used in SCC modeling: (i) Continuity Equation: The continuity equation expresses the conservation of mass and is fundamental to fluid flow modeling. In the case of SCC, it can be written as: (1) $$\mathit{\partial }\rho /\partial \mathit{t}+\nabla ·\left(\rho \mathit{u}\right)=0$$(1) where ρ is the density of SCC, t is time, u is the velocity vector, and ∇ · represents the divergence operator. (ii) Navier-Stokes Equations: The Navier-Stokes equations describe the conservation of momentum and are used to model fluid flow. They can be written as: (2) $$\mathit{\rho }(\partial \mathit{u}/\partial \mathit{t}+u·\nabla \mathit{u})=-\nabla \mathit{p}+\nabla ·\mathit{\tau }+\mathit{\rho }\mathit{g}$$(2) where p is the pressure, τ is the stress tensor, and g is the acceleration due to gravity. (iii) Bingham Plastic Model: SCC exhibits non-Newtonian behavior, and the Bingham plastic model is often used to describe its rheological properties. The Bingham model relates the shear stress (τ) to the shear rate (du/dy) and yield stress (τ₀) as follows: (3) $$\mathit{\tau }={\mathit{\tau }}_0+\mathit{\mu }\left(\mathit{d}\mathit{u}/\mathit{d}\mathit{y}\right)$$(3) where μ is the plastic viscosity. (iv) Constitutive Equations: Constitutive equations describe the relationship between stress and strain in the material. Various models, such as the Herschel-Bulkley model or the power-law model, can be used to represent the rheological behavior of SCC. (v) Particle Interaction: In Smoothed Particle Hydrodynamics (SPH) methods, the interaction between particles is typically modeled using a smoothing kernel function. Commonly used kernel functions include the cubic spline kernel or the Gaussian kernel. (vi) Boundary Conditions: Boundary conditions specify the behavior of SCC at the boundaries of the simulation domain. These can include velocity or pressure boundary conditions, as well as any external forces acting on the system. (vi) Coupling Equations: When combining ANSYS and SPH for SCC modeling, coupling equations are used to exchange information between the finite element solver and the SPH solver (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). These equations ensure the correct interaction between the fluid flow simulation and the SPH particle representation. It's important to note that the mathematics involved in SCC modeling can be quite complex, and the specific equations and models used may vary depending on the desired level of accuracy and the specific characteristics of the SCC being studied. Researchers and engineers working in this field often develop and refine mathematical models based on experimental data and empirical correlations to accurately represent the behavior of SCC during flow.

2.7. Continuity equations of multiphysical flow of self-compacting concrete

The continuity equation for the multiphysical flow of self-compacting concrete (SCC) describes the conservation of mass. It ensures that the mass of SCC is conserved during the flow process. The continuity equation can be expressed mathematically as follows: (4) $$\partial \mathit{\rho }/\partial \mathit{t}+\nabla ·\left(\mathit{\rho }\mathit{v}\right)=0$$(4) where:

• ∂ρ/∂t is the partial derivative of density ρ with respect to time t, representing the rate of change of density with time.

• ∇ is the del operator, representing the divergence operator.

• · denotes the dot product.

• ρ is the density of SCC.

• v is the velocity vector of the SCC flow.

In this equation, the first term on the left side (∂ρ/∂t) represents the temporal change in density. The second term on the left side (∇ · (ρv)) represents the spatial change in density due to the divergence of the velocity field. It accounts for the fact that the mass of SCC can change due to variations in flow velocity. The continuity equation states that the sum of these temporal and spatial changes in density must be zero, implying that mass is conserved. In other words, the rate of change of density at any point in the SCC flow field is balanced by the inflow and outflow of mass through the boundaries of the control volume (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). Solving the continuity equation, along with other governing equations such as the momentum equations and constitutive equations, provides a mathematical framework to model and simulate the multiphysical flow behavior of SCC.

2.8. Navier-Stokes equations of multiphysical flow of self-compacting concrete

The Navier-Stokes equations describe the conservation of momentum in fluid flow and are widely used to model the multiphysical flow of self-compacting concrete (SCC). The Navier-Stokes equations can be written as follows:

Continuity Equation: (5) $$\partial \mathit{\rho }/\partial \mathit{t}+\nabla ·\left(\mathit{\rho }\mathit{v}\right)=0$$(5)

Momentum Equations: (6) $$\partial \left(\text{ρv}\right)/\partial t+\nabla ·({\text{ρv}}^{⃗}\otimes v^{⃗})=-\nabla p+\nabla ·\text{τ}+\text{ρ}\mathit{g}$$(6) where:

• ∂ρ/∂t is the partial derivative of density ρ with respect to time t.

• ∂(ρv)/∂t is the partial derivative of the momentum density ρv with respect to time t.

• ∇ is the del operator, representing the divergence operator.

• · denotes the dot product.

• v^→ is the velocity vector of the SCC flow.

• p is the pressure.

• τ is the stress tensor.

• g is the acceleration due to gravity.

In the momentum equations, the first term on the left side (∂(ρv)/∂t) represents the temporal change in momentum density. The second term on the left side (∇ · (ρv^→⊗ v^→)) represents the spatial change in momentum density due to the divergence of the convective flux term (ρv^→⊗ v^→), where ⊗ denotes the outer product. This term accounts for the transport of momentum in the flow (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). On the right side of the momentum equations, the term −∇p represents the pressure gradient, which drives the flow. The term ∇ · τ represents the divergence of the stress tensor, accounting for the viscous forces in the flow. The final term ρg represents the gravitational force acting on the SCC. Solving the Navier-Stokes equations, along with appropriate constitutive equations for the rheological behavior of SCC, provides a mathematical framework to simulate and analyze the multiphysical flow of self-compacting concrete.

2.9. Bingham plastic model equations of multiphysical flow of self-compacting concrete

The Bingham plastic model is commonly used to describe the flow behavior of self-compacting concrete (SCC). SCC is a complex material that exhibits both fluid-like and solid-like behavior, and the Bingham plastic model provides a simplified representation of its flow properties (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The model assumes that the material behaves as a rigid solid until a certain yield stress is exceeded, at which point it starts to flow like a viscous fluid. The Bingham plastic model for SCC can be expressed mathematically using the following equations:

(i) Yield Stress Equation: (7) $$\mathit{\tau }={\mathit{\tau }}_0+\text{μ·γ}$$(7)

In this equation, τ is the shear stress, τ₀ is the yield stress (the stress required to initiate flow), μ is the plastic viscosity (a measure of the material's resistance to flow), and γ is the shear rate (the rate at which the material is deforming).

(ii) Flow Rule Equation: (8) $$\mathit{\tau }=\text{μ·γ}$$(8)

This equation represents the flow behavior of the SCC once the yield stress has been exceeded. It states that the shear stress is directly proportional to the shear rate, with μ representing the plastic viscosity. These equations can be used to simulate and analyze the flow behavior of SCC in different scenarios. It is important to note that the values of yield stress (τ₀) and plastic viscosity (μ) are material-dependent and can be determined through experimental testing.

2.10. Particle interaction equations of multiphysical flow of self-compacting concrete

Smoothed Particle Hydrodynamics (SPH) is a meshless numerical method commonly used to simulate multiphysical flows, including the flow of self-compacting concrete (SCC). SPH represents the material as a collection of particles and uses interpolation techniques to compute the properties and interactions between these particles. When simulating SCC, particle interactions play a crucial role in capturing the complex behavior of the material. While SPH doesn't have explicit equations for particle interactions, the method incorporates the following concepts to model them: (i) Kernel Function: The kernel function is a smoothing function that defines the influence of each particle on its neighbors. It determines how the properties of neighboring particles are interpolated to compute the values at a specific point (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). Commonly used kernel functions include the cubic spline and the Gaussian function. (ii) Interparticle Forces: Interparticle forces are used to model the interactions between particles in SCC. Various forces can be considered, such as repulsive forces to capture particle collisions, cohesive forces to represent particle bonding, and viscous forces to account for the viscosity of the material. The nature and formulation of these forces depend on the specific behavior being simulated. (iii) Material Constitutive Models: Constitutive models are used to describe the behavior of SCC under different loading conditions (Onyelowe, Kontoni, Onyia, et al., 2023). These models define the relationships between stress, strain, and other material properties. In SPH, constitutive models are typically incorporated into the computation of interparticle forces to capture the mechanical response of the material. (iv) Boundary Conditions: Boundary conditions are imposed to represent the external constraints on the SCC flow. They can include fixed boundaries, moving boundaries, and interaction with other materials or structures. These conditions influence the behavior of particles near the boundaries and are essential for accurately simulating real-world scenarios. By combining these concepts, SPH simulations can capture the interactions between particles in SCC, allowing for the prediction of flow behavior, segregation, and other phenomena. It's worth noting that the specific implementation and formulation of SPH for SCC may vary depending on the research or simulation software being used.

2.11. Boundary conditions equations of multiphysical flow of self-compacting concrete

When simulating the multiphysical flow of self-compacting concrete (SCC), boundary conditions are essential to model the interactions between the SCC material and its surroundings. The choice and formulation of boundary conditions depend on the specific analysis being performed (Kaveh et al., 2018; Onyelowe, Kontoni, Onyia, et al., 2023). Here are some common boundary conditions used in SCC simulations: (i) Inflow Boundary: An inflow boundary condition is applied at the location where SCC is introduced into the computational domain. The inflow condition specifies the properties of the incoming SCC material, such as velocity, temperature, and concentration of constituents. (ii) Outflow Boundary: An outflow boundary condition is applied at the location where SCC exits the computational domain (Kaveh et al., 2018; Onyelowe et al., 2022a; Onyelowe, Kontoni, Onyia, et al., 2023; Onyelowe, Naghizadeh, et al., 2023). The outflow condition can be either a free outflow condition, where the flow properties are allowed to adjust freely, or a specified outflow condition, where certain properties, such as pressure or velocity, are prescribed. (iii) Wall Boundary: A wall boundary condition is used to represent solid surfaces or boundaries that interact with the SCC material. At a wall boundary, the velocity of the SCC material is typically set to zero, representing a no-slip condition. Depending on the specific analysis, additional conditions such as temperature or concentration can be specified at the wall (Sigalotti et al., 2021). (iv) Symmetry Boundary: A symmetry boundary condition is applied when the system exhibits symmetry and only a portion of the domain needs to be simulated (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The symmetry condition imposes symmetry in the flow properties across the boundary, such as velocity or pressure. (v) Interface Boundary: In cases where SCC interacts with other materials or phases, such as when it flows into a reinforcing structure or encounters air or water, interface boundary conditions are used (Hosseinpoor et al., 2017). These conditions consider the exchange of properties, such as momentum, mass, or heat, across the interface. (vi) Moving Boundary: In some simulations, boundaries can be dynamic or moving, such as when modeling the filling of a formwork or the flow through a rotating mixer. Moving boundary conditions are applied to account for the changing shape or position of the boundary as the simulation progresses. The specific equations used to implement these boundary conditions depend on the numerical method and software being employed for the simulation. Different techniques, such as interpolation, extrapolation, or direct specification, can be used to enforce the desired boundary conditions for each property of interest.

2.12. Coupling of ANSYS and SPH

Coupling ANSYS and SPH (Smoothed Particle Hydrodynamics) for the simulation of multiphysical flow, such as self-compacting concrete, can be a complex task. While I can provide you with a general overview, it's important to note that the specific implementation details may vary depending on the software versions and your simulation requirements. Here's a high-level approach to coupling ANSYS and SPH: (i) Preprocessing: (a) Create the geometry of the domain and define the boundary conditions in ANSYS. (b) Define the material properties and constitutive models for self-compacting concrete in ANSYS. (c) Generate the SPH particles that represent the concrete mixture and its behavior using an SPH software package. (ii) Data exchange: (a) Develop a data exchange interface between ANSYS and the SPH solver. This can involve writing custom scripts or using available coupling tools. (b) Transfer the initial geometry, boundary conditions, and material properties from ANSYS to the SPH solver. (iii) Simulation: (a) Perform the SPH simulation using the transferred data. The SPH solver will simulate the fluid-like behavior of the self-compacting concrete mixture, including its flow, deformation, and interaction with other objects or boundaries. (b) Extract the relevant simulation results from the SPH solver, such as velocity, pressure, and stress distribution. (iv) Postprocessing: (a) Transfer the simulation results back from the SPH solver to ANSYS. (b) Analyze and visualize the results in ANSYS to obtain insights into the multiphysical behavior of the self-compacting concrete. (c) Perform further postprocessing tasks, such as computing forces, displacements, or any other quantities of interest (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). It's important to note that the coupling process often requires close collaboration between experts in both ANSYS and SPH software, as well as a solid understanding of the underlying physics and numerical methods involved. Additionally, there may be commercial or open-source software available that provides pre-built coupling capabilities, which can simplify the implementation process. Please keep in mind that the information provided here is a general guideline, and it's recommended to consult the documentation and resources specific to the ANSYS and SPH software versions you are using for more detailed instructions and support.

2.13. Modeling free surface flow

Modeling free surface flow equations involves describing the behavior of fluid flow on the surface of a body of water or a fluid flow with a free surface (Toosi et al., 2015). There are various mathematical models that can be used to represent free surface flow, and two commonly used approaches are the shallow water equations and the Navier-Stokes equations. (i) Shallow Water Equations: The shallow water equations are a simplified form of the Navier-Stokes equations that are commonly used for modeling free surface flow in open channels and large-scale flows (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Toosi et al., 2015; Wang et al., 2016). They assume that the flow depth is much smaller compared to the horizontal length scale, and the velocity varies primarily in the horizontal direction. The one-dimensional shallow water equations in the conservative form are:

Continuity equation: ∂h/∂t + ∂(hu)/∂x = 0, Momentum equation: ∂(hu)/∂t + ∂(hu² + gh²/2)/∂x = −g(h + ζ)∂ζ/∂x; where h is the flow depth, u is the velocity in the x-direction, t is time, x is the horizontal direction, g is the acceleration due to gravity, and ζ represents the free surface elevation. These equations can be solved numerically using finite difference, finite volume, or finite element methods, along with appropriate boundary conditions. (ii) Navier-Stokes Equations: The Navier-Stokes equations are the fundamental equations for fluid flow and can be used to model free surface flow as well. However, solving the full Navier-Stokes equations is computationally expensive and often requires simplifications and assumptions (Onyelowe, Kontoni, Onyia, et al., 2023). The three-dimensional Navier-Stokes equations in an incompressible form are Continuity equation: ∇⋅v = 0, Momentum equation: ∂v/∂t + (v⋅∇)v = −∇P/ρ + ν∇²v + g; where v is the velocity vector, P is the pressure, ρ is the fluid density, ν is the kinematic viscosity, and g represents any external forces such as gravity. To model free surface flow using the Navier-Stokes equations, additional equations or models are needed to track the position of the free surface (Toosi et al., 2015). These can include the volume of fluid (VOF) method, level set method, or surface capturing techniques. Numerical methods such as finite volume, finite element, or finite difference methods are commonly used to discretize and solve the Navier-Stokes equations along with the additional free surface tracking equations. In practice, the choice between the shallow water equations and the Navier-Stokes equations depends on the scale and complexity of the flow being modeled and the available computational resources.

3. Theory and formulation

3.1. Mathematics of the SPH

Smoothed Particle Hydrodynamics (SPH) is a computational method used for simulating fluid dynamics and other physical phenomena. It is commonly employed in areas such as astrophysics, computational fluid dynamics, and computer graphics. SPH represents a fluid as a collection of particles that interact with each other through a kernel function. In SPH, the state variables (such as density, pressure, velocity, etc.) are associated with each particle (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). To compute the values of these variables at any point in space, a smoothing operation is performed using the neighboring particles. The key idea is that each particle contributes to the values of its neighbors through a kernel function, which assigns weights to the neighboring particles based on their distance from the target point. Let's consider a scalar quantity, such as density, as an example. The density at a given point in space is computed by summing the contributions from nearby particles as presented in Figure 1.

Figure 1. The SPH problem domain of continuous particles in SCC.

The SPH density estimator is defined as: (9) $$\mathit{\rho }\left(\mathit{x}\right)=\sum \left({\mathit{m}}_{\mathit{j}}*\mathit{W}\left(\left|\mathit{x}-{\mathit{x}}_{\mathit{j}}\right|,\mathit{h}\right)\right)$$(9) where:

• ρ(x) is the density at point x,

• m_j is the mass of the jth particle,

• x_j is the position of the jth particle,

• W is the kernel function,

• |x − x_j| is the distance between the target point x and the jth particle,

• h is the smoothing length that determines the range of influence of each particle.

The kernel function W(|x − x_j|, h) assigns weights to the neighboring particles based on their distance from x. Commonly used kernel functions in SPH include the Gaussian kernel and the cubic spline kernel. These functions satisfy certain properties, such as normalization and compact support, to ensure accurate and stable simulations. Once the density is computed, other quantities, such as pressure and velocity, can be obtained by solving appropriate equations. For example, the pressure can be computed using an equation of state that relates the density and pressure of the fluid. The velocity of a particle can be updated by considering the pressure and viscous forces acting on it. In addition to the density, other physical quantities are also estimated using similar SPH formulations. The key idea is to compute the values of these quantities by summing the contributions from nearby particles using appropriate kernel functions. It's worth noting that SPH is a meshless method, which means it does not rely on a fixed grid or mesh to discretize the computational domain. Instead, it dynamically adjusts the distribution of particles based on the density and other properties of the fluid (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The mathematics of SPH can become quite involved, especially when considering complex physical phenomena and their governing equations. The above explanation provides a simplified overview of the basic principles behind SPH, but there are more advanced techniques and refinements that researchers have developed to improve the accuracy and efficiency of SPH simulations.

3.2. Formulation of SPH for fluid flow

The smoothed particle hydrodynamics (SPH) method for fluid flow involves the discretization of the fluid into a set of particles that interact with each other. Each particle carries certain properties such as position, velocity, and density. The governing equations of fluid dynamics, such as the continuity equation and the Navier-Stokes equations, are solved numerically using the SPH formulation. Here is a general formulation of SPH for fluid flow: (i) Particle Discretization: (a) The fluid domain is discretized into a set of particles, each representing a small region of the fluid. (b) Each particle has properties, such as position (x), velocity (v), density (ρ), and pressure (P). (c) The particles are typically distributed in a way that approximates the initial fluid configuration. (ii) Smoothing Length: (a) A smoothing length parameter (h) is defined for each particle, which determines the range of influence of the particle. (b) The smoothing length regulates the spatial resolution of the simulation and affects the accuracy and stability of the SPH method. (iii) Kernel Function: (a) A kernel function (W) is used to interpolate values between particles and determine their influence on each other. (b) The kernel function assigns weights to neighboring particles based on their distance from a target particle. (c) Commonly used kernel functions include the Gaussian kernel and the cubic spline kernel. (iv) Density Estimation: (a) The density at each particle is estimated by summing the contributions from neighboring particles using the kernel function. (b) The density estimation equation is typically defined as: (10) $${\mathit{\rho }}_{\mathit{i}}=\sum \left({\mathit{m}}_{\mathit{j}}*\mathit{W}\left(\left|{\mathit{x}}_{\mathit{i}}-{\text{x}}_{\mathit{j}}\right|,\mathit{h}\right)\right)$$(10) where ρ_i is the density of particle i, m_j is the mass of particle j, and |x_i − x_j| is the distance between particles i and j. (v) Pressure Calculation: (a) The pressure at each particle is computed using an equation of state that relates the density and pressure of the fluid. (b) The equation of state may vary depending on the specific fluid being simulated (e.g. ideal gas law for compressible fluids). (vi) Force Computation: (a) The forces acting on each particle, including pressure forces and viscous forces, are computed. (b) The pressure force can be obtained from the pressure gradient, and the viscous force is often modeled using the Navier-Stokes equations. (vii) Time Integration: (a) The equations of motion, including the continuity equation and the Navier-Stokes equations, are integrated over time to update the particle positions and velocities. (b) This is typically done using numerical integration schemes such as the Verlet method or the Runge-Kutta method. (viii) Boundary Conditions: (a) Appropriate boundary conditions are imposed to simulate the interactions between the fluid and its boundaries. (b) Examples of boundary conditions include no-slip conditions at solid walls or inflow/outflow conditions at open boundaries. By iteratively updating the particle properties based on the computed forces and integrating the equations of motion over time, the SPH method simulates the behavior of the fluid flow (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The accuracy and stability of the simulation depend on factors such as the choice of kernel function, the smoothing length, and the numerical integration scheme used. Various enhancements and modifications to the basic SPH formulation have been proposed to address specific challenges and improve the accuracy of fluid flow simulations.

3.3. Formulation of SPH for self-compacting concrete flow time

The smoothed particle hydrodynamics (SPH) method can also be applied to simulate the flow behavior of self-compacting concrete (SCC). SCC is a highly flowable concrete mix that can fill complex molds and structures under its own weight without the need for external compaction. The SPH formulation for SCC flow time involves considering the rheological properties of the concrete and modeling its behavior using appropriate constitutive equations. Here is a general formulation of SPH for SCC flow time: (i) Particle Discretization: (a) The SCC domain is discretized into a set of particles, each representing a small volume of the concrete. (b) Each particle carries properties such as position (x), velocity (v), density (ρ), and other relevant parameters specific to SCC (e.g. slump flow). (ii) Smoothing Length: (a) A smoothing length parameter (h) is defined for each particle, determining the range of influence of the particle. (b) The smoothing length affects the spatial resolution of the simulation and must be chosen appropriately for accurate results. (iii) Kernel Function: (a) A kernel function (W) is used to interpolate values between particles and determine their interaction. (b) The kernel function assigns weights to neighboring particles based on their distance from a target particle. (c) Common kernel functions used in SPH, such as the Gaussian or cubic spline kernel, can be employed for SCC simulations. (iv) Rheological Model: (a) A rheological model is employed to describe the flow behavior of SCC. (b) The chosen model should capture the complex properties of self-compacting concrete, such as yield stress, plastic viscosity, and thixotropy. (c) Various rheological models have been proposed for SCC, including the Bingham model, Herschel-Bulkley model, and others. (v) Constitutive Equations: (a) The constitutive equations of the selected rheological model are used to relate stress to strain rate in the SCC flow. (b) These equations describe the flow behavior of the concrete under different conditions, such as shear flow or extensional flow. (c) The constitutive equations may involve parameters specific to the SCC mix, which are determined experimentally. (vi) Force Computation: (a) The forces acting on each particle are calculated based on the rheological model and the constitutive equations. (b) These forces include shear forces, viscosity forces, and other relevant forces affecting the flow of SCC. (c) The forces are computed by considering the interactions between neighboring particles using the kernel function. (vii) Time Integration: (a) The equations of motion, derived from the rheological model and the forces calculated, are integrated over time. (b) Time integration schemes, such as the Verlet method or the Runge-Kutta method, can be employed to update particle positions and velocities. (viii) Boundary Conditions: (a) Appropriate boundary conditions are imposed to simulate the interactions between SCC and its surroundings. (b) These conditions may include no-slip conditions at solid walls, inflow/outflow conditions at openings, or other relevant constraints. By simulating the particle interactions, forces, and time integration, the SPH method can provide insights into the flow behavior and flow time of self-compacting concrete (Onyelowe, Kontoni, Onyia, et al., 2023). It is important to note that the accuracy of the simulation depends on the chosen rheological model, the parameters used, and the validation against experimental data.

3.4. Formulation of ANSYS for self-compacting concrete flow time

To simulate the flow time of self-compacting concrete (SCC) using ANSYS, you can follow a general procedure that involves defining the material properties, setting up the geometry, applying boundary conditions, and solving the fluid flow problem. Here's a step-by-step formulation guide: (i) Material properties: Define the properties of SCC, such as density, viscosity, and yield stress. These properties may vary depending on the specific SCC mix design. (ii) Geometry: Create a 3D model of the geometry representing the structure or element where SCC flow is being analyzed. This can be a simple geometry, such as a column or beam, or a more complex structure. (iii) Boundary conditions: (a) Define the inlet and outlet boundaries for the SCC flow. Typically, the inlet is where the SCC is introduced, and the outlet is where it exits or reaches a desired state. (b) Specify the boundary conditions, such as pressure or velocity at the inlet, and pressure or zero velocity at the outlet. (iv) Fluid flow analysis: (a) Set up a transient fluid flow analysis to simulate the time-dependent behavior of SCC flow. (b) Define the governing equations for fluid flow, such as the Navier-Stokes equations, using the appropriate fluid flow solver in ANSYS (e.g. Fluent or CFX). (c) Apply the specified material properties and boundary conditions to the fluid flow analysis. (v) Meshing: Create a mesh for the geometry to discretize it into smaller elements. The mesh should be fine enough to capture the flow behavior accurately. Consider using a structured or unstructured mesh depending on the complexity of the geometry. (vi) Solver settings: Configure the solver settings, such as time step size, convergence criteria, and solution method, based on your specific needs and constraints. (vii) Solve the problem: (a) Run the simulation to solve the fluid flow problem using the ANSYS solver. (b) Monitor the convergence of the solution and adjust solver settings if necessary. (viii) Post-processing: (a) Analyze the results obtained from the simulation, such as flow velocities, pressure distributions, and flow time. (b) Visualize the results using ANSYS post-processing tools or export the data for further analysis. It is to be noted that the specific details and steps may vary depending on the version of ANSYS you are using and the complexity of the SCC flow problem. It's recommended to consult the ANSYS documentation or contact ANSYS support for more detailed guidance on using their software for SCC flow analysis (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016).

3.5. Mathematics of ANSYS for fluid flow

ANSYS is a general-purpose finite element analysis software that offers several tools and solvers for fluid flow analysis, such as ANSYS Fluent and ANSYS CFX. The mathematics underlying ANSYS for fluid flow simulations are based on the fundamental principles of fluid mechanics and the governing equations for fluid flow. The governing equations for fluid flow are typically the Navier-Stokes equations, which describe the conservation of mass, momentum, and energy in a fluid. The Navier-Stokes equations can be written as follows:

(i) Conservation of mass (continuity equation): (11) $$\nabla ·\left(\mathit{\rho }\mathit{V}\right)=0$$(11) where ∇ is the del operator, ρ is the density of the fluid, and V is the velocity vector.

(ii) Conservation of momentum (Navier-Stokes equations): (12) $$\mathit{\rho }\left(\mathit{d}\mathit{V}/\mathit{d}\mathit{t}\right)=-\nabla \mathit{P}+\nabla ·(\mathit{\mu }(\nabla \mathit{V}+{(\nabla \mathit{V})}^{\mathit{T}}))+\mathit{\rho }\mathit{g}$$(12) where dV/dt is the material derivative of velocity, P is the pressure, μ is the dynamic viscosity of the fluid, (∇V + (∇V)^T) is the rate-of-strain tensor, and g is the acceleration due to gravity. In addition to the Navier-Stokes equations, ANSYS may incorporate other equations or models depending on the specific solver and analysis being performed. These additional equations can account for various physical phenomena, such as turbulence, multiphase flows, heat transfer, and chemical reactions. To solve the governing equations, ANSYS uses numerical methods, typically finite element or finite volume methods, to discretize the domain into smaller elements or control volumes. The equations are then solved iteratively over time or until a convergence criterion is met. During the solution process, ANSYS employs algorithms to handle the nonlinearities and coupling between the different equations. These algorithms may include techniques such as linearization, iterative solvers, and turbulence models. The accuracy and efficiency of ANSYS simulations can be influenced by factors such as the mesh quality, time step size, convergence criteria, solver settings, and material models chosen. Therefore, careful consideration of these factors is necessary to obtain accurate and reliable results. It's important to note that the specific mathematical details and numerical methods used by ANSYS can vary depending on the solver and analysis being performed (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). ANSYS documentation and resources provide more in-depth information on the mathematics and algorithms employed by each specific solver.

3.6. Conventional SPH formulation

Conventional SPH (Smoothed Particle Hydrodynamics) is a meshless numerical method commonly used for simulating fluid flows. It was first introduced by Gingold and Monaghan in 1977 and has since been widely applied in various fields, including astrophysics, computational fluid dynamics (CFD), and computer graphics (Hosseinpoor et al., 2017; Onyelowe, Kontoni, Onyia, et al., 2023; Sigalotti et al., 2021; Wang et al., 2016). The basic idea behind SPH is to represent a fluid as a set of particles, where each particle carries certain properties such as position, velocity, density, and pressure. The fluid is discretized into these particles, and the governing equations of fluid dynamics are approximated using interpolation and smoothing techniques. The conventional SPH formulation involves several key steps: (i) Particle Discretization: The fluid domain is discretized into a set of particles. These particles are typically distributed throughout the fluid domain, and their positions are tracked throughout the simulation. (ii) Smoothing Kernel: A smoothing kernel function is defined to interpolate properties between neighboring particles. The kernel function determines the influence of each particle on its neighbors and is usually radially symmetric. A popular choice is the cubic spline kernel. (iii) Interpolation: The properties of a particle are interpolated from its neighboring particles using the smoothing kernel. This interpolation is used to approximate the derivatives of the properties, such as velocity and density gradients. (iv) Density Estimation: The density of each particle is estimated based on the mass distribution of neighboring particles. This step involves summing the masses of nearby particles weighted by the smoothing kernel. (v) Pressure Calculation: The pressure at each particle is calculated based on an equation of state or other pressure models. The pressure is typically derived from the equation of continuity (mass conservation) and the equation of state. (vi) Forces Calculation: Forces acting on each particle, such as pressure forces, viscous forces, and external forces (e.g. gravity), are computed based on the properties of neighboring particles. These forces influence the motion of the particles. (vii) Time Integration: The equations of motion are numerically integrated in time to update the particle positions and velocities. Various time integration schemes, such as the leapfrog or Verlet method, can be used. (viii) Iteration: The simulation advances in time by iteratively repeating steps (iii)–(vii) until the desired simulation time is reached. Conventional SPH has been successful in simulating a wide range of fluid phenomena, including free-surface flows, multiphase flows, and complex fluid interactions (Onyelowe, Kontoni, Onyia, et al., 2023). However, it also has some limitations, such as difficulties in handling sharp interfaces and numerical instabilities. Various modifications and enhancements to the conventional SPH formulation have been proposed to address these limitations.

3.7. SPH particle approximation

In Smoothed Particle Hydrodynamics (SPH), particles are used to discretize the fluid domain and approximate its properties. The distribution of these particles represents the fluid and their properties, such as position, velocity, density, and pressure, are used to simulate the fluid behavior. The approximation of fluid properties is done through interpolation using a smoothing kernel function. Here's an overview of the SPH particle approximation: (i) Particle Discretization: The fluid domain is divided into a set of particles. The particles can be distributed uniformly or non-uniformly within the domain, depending on the specific requirements of the simulation. (ii) Smoothing Kernel: A smoothing kernel function is defined to determine the influence of neighboring particles on each particle (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The kernel function is usually radially symmetric and decreases with distance from a particle. It assigns weights to neighboring particles based on their proximity to the target particle. (iii) Interpolation: The properties of a particle are interpolated from its neighboring particles using the smoothing kernel. The kernel is evaluated for each neighboring particle, and their contributions are summed up to calculate the interpolated value at the target particle. This interpolation step is crucial for approximating derivatives and gradients of fluid properties. (iv) Density Estimation: The density of a particle is estimated by summing the masses of neighboring particles weighted by the smoothing kernel. The kernel is used to determine the influence of each neighboring particle on the density of the target particle (Onyelowe, Kontoni, Onyia, et al., 2023). This step ensures that the density is properly computed based on the distribution of particles in the vicinity. (v) Pressure Calculation: The pressure at each particle is calculated based on an equation of state or other pressure models. The pressure is typically derived from the equation of continuity (mass conservation) and the equation of state, which relates the pressure to the density and other fluid properties. (vi) Force Calculation: Forces acting on each particle, such as pressure forces, viscous forces, and external forces (e.g. gravity), are computed based on the properties of neighboring particles (Onyelowe, Kontoni, Onyia, et al., 2023). The smoothing kernel is again used to determine the influence of neighboring particles on the force calculation. (vii) Time Integration: The equations of motion are numerically integrated in time to update the particle positions and velocities. Time integration schemes, such as the leapfrog or Verlet method, are utilized to advance the simulation in time. The forces calculated in the previous step are used to update the particle velocities and positions. By employing these steps, SPH approximates the fluid properties and behavior based on the interactions between particles. The use of the smoothing kernel allows for a continuous representation of fluid properties, enabling the simulation of complex fluid phenomena.

3.8. SPH kernel function

In Smoothed Particle Hydrodynamics (SPH), a key component is the smoothing kernel function. The kernel function is used to interpolate properties between neighboring particles, determining their influence on each other. The choice of kernel function affects the accuracy and stability of the SPH simulation. Here are some commonly used SPH kernel functions: (i) Gaussian Kernel: The Gaussian kernel is widely used in SPH simulations. It is defined as: (13) $$\mathit{W}\left(\mathit{r},\mathit{h}\right)=\left(1/\left(\mathit{\pi }*{\mathit{h}}^2\right)\right)*\mathit{e}\mathit{x}\mathit{p}\left(-{\left(\mathit{r}/\mathit{h}\right)}^2\right)$$(13) where r is the distance between particles, and h is the smoothing length or kernel radius. The Gaussian kernel has compact support, meaning it decays rapidly as the distance between particles increases. (ii) Cubic Spline Kernel: The cubic spline kernel is another commonly used kernel function. It is given by: (14) $$\mathit{W}\left(\mathit{r},\mathit{h}\right)=\left(10/\left(7*\mathit{\pi }*{\mathit{h}}^2\right)\right)*{\left(1-\mathit{q}\right)}^3$$(14) where q = r/h is the normalized distance between particles. The cubic spline kernel is smooth and has continuous derivatives up to the second order. (iii) Quintic Spline Kernel: The quintic spline kernel provides higher order continuity compared to the cubic spline kernel. It is defined as: (15) $$\mathit{W}\left(\mathit{r},\text{h}\right)=\left(21/\left(16*\mathit{\pi }*{\text{h}}^3\right)\right)*{\left(1-\mathit{q}\right)}^4*\left(1+\text{4}*\mathit{q}\right)$$(15)

The quintic spline kernel is useful for simulating highly viscous fluids or capturing fine details in the flow. (iv) Wendland C2 Kernel: The Wendland C2 kernel is a compact support kernel that ensures higher order continuity. It is given by: (16) $$\mathit{W}\left(\mathit{r},\mathit{h}\right)={\left(\mathit{C}/\mathit{h}\right)}^{\mathit{d}}*{\left(1-\mathit{q}\right)}^4*\left(4*\mathit{q}+\text{1}\right)$$(16) where C is a normalization constant, d is the dimensionality of the problem (2 for 2D, 3 for 3D), and q = r/h. The Wendland C2 kernel is known for its smoothness and good stability properties. These are just a few examples of SPH kernel functions used in practice. Each kernel has its own characteristics and suitability for different types of simulations (Hosseinpoor et al., 2017; Onyelowe, Kontoni, Onyia, et al., 2023; Sigalotti et al., 2021; Wang et al., 2016). The choice of kernel should be based on the specific requirements of the problem, such as the desired smoothness, accuracy, and computational efficiency.

3.9. SPH kernel gradient correction

In computational physics and astrophysics, the Smoothed Particle Hydrodynamics (SPH) method is a numerical technique used for simulating fluid dynamics. The SPH method represents a fluid as a set of particles, and the properties of the fluid are calculated based on the properties of these particles (Onyelowe, Kontoni, Onyia, et al., 2023). One challenge in using SPH is accurately calculating the gradients of the fluid properties, such as density or pressure. The gradients are used to determine the forces acting on the particles and to interpolate properties between neighboring particles. In the original SPH formulation, the gradients are estimated using a simple kernel function, such as the cubic spline kernel. However, the standard SPH formulation can suffer from an issue known as the ‘particle disorder’ problem. This problem arises because particles in SPH simulations are not uniformly distributed, and there can be irregularities in the particle distribution. As a result, the gradient estimates based on the kernel function alone can be inaccurate, leading to spurious numerical artifacts and instabilities in the simulation. To address this problem, several gradient correction techniques have been proposed in the literature. These techniques aim to improve the accuracy of gradient estimates in SPH simulations, particularly in regions with particle disorder. One common approach is to include additional terms in the gradient estimation formula that take into account the local particle distribution (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). These terms can be derived from the derivative of the kernel function or by fitting the gradient to a smoother function that depends on the particle distribution. The choice of gradient correction technique depends on the specific problem being simulated and the desired accuracy. Some commonly used gradient correction methods include: (i) Pressure-Entropy SPH (PESPH): PESPH introduces an additional term in the gradient estimation formula that depends on the gradient of the entropy. This method helps to suppress spurious pressure oscillations and improves the accuracy of the pressure gradient estimation. (ii) Density-Gradient SPH (DGSPH): DGSPH corrects the density gradient estimates by fitting them to a smoother function derived from the particle distribution. This method improves the accuracy of density interpolation and reduces particle disorder effects. (iii) Kernel Gradient Estimator (KGE): KGE modifies the standard kernel function by including additional terms that depend on the particle distribution (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). These terms improve the accuracy of the gradient estimation and help to mitigate numerical artifacts. These are just a few examples of gradient correction techniques used in SPH simulations. The choice of method depends on the specific problem and the desired level of accuracy. Researchers continue to develop and refine gradient correction techniques to improve the accuracy and stability of SPH simulations (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016).

The gradient correction equations in Smoothed Particle Hydrodynamics (SPH) aim to improve the accuracy of gradient estimates by taking into account the irregular distribution of particles. There are different approaches to gradient correction, and I will provide you with a few examples of commonly used equations. Note that these equations are just a few possibilities, and there are various other methods and formulations available in the literature. (i) Pressure-Entropy SPH (PESPH): In PESPH, an additional term is added to the gradient estimation formula based on the gradient of the entropy. The corrected gradient equation for a quantity, let's say, A, can be expressed as: (17) $$\nabla {\mathit{A}}_{\mathit{i}}={\sum }_{\mathit{j}}\left({\mathit{A}}_{\mathit{j}}-{\text{A}}_{\mathit{i}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}+\left({\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\rho }_{\mathit{j}}\right){\nabla }_{\mathit{i}}{\mathit{A}}_{\mathit{j}}\right)*{\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}$$(17) where i and j represent the target particle and its neighboring particles, respectively. A_i and ρ_i represent the value of quantity A and density at particle i, m_j represents the mass of particle j, and W_ij is the kernel function. (ii) Density-Gradient SPH (DGSPH):DGSPH corrects the density gradient estimates by fitting them to a smoother function derived from the particle distribution. The corrected gradient equation for density (ρ) can be written as: (18) $$\nabla {\mathit{\rho }}_{\mathit{i}}={\sum }_{\mathit{j}}\left({\mathit{\rho }}_{\mathit{j}}-{\mathit{\rho }}_{\mathit{i}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}+\left({\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\mathit{\rho }}_{\mathit{j}}\right){\nabla }_{\mathit{i}}{\mathit{\rho }}_{\mathit{j}}\right)*{\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}$$(18)

Here, the first term represents the standard SPH density gradient estimation, and the second term is the correction term that depends on the gradient of density. (iii) Kernel Gradient Estimator (KGE):KGE modifies the kernel function to include additional terms that depend on the particle distribution. One example of a corrected gradient equation using KGE is: (19) $$\nabla {\mathit{A}}_{\mathit{i}}={\sum }_{\mathit{j}}\left({\mathit{A}}_{\mathit{j}}-{\text{A}}_{\mathit{i}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}+\left({\sum }_{\mathit{j}}\left({\mathit{A}}_{\mathit{j}}-{\text{A}}_{\mathit{i}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}\right)*{\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}$$(19)

This equation includes an additional term that multiplies the standard gradient estimate by the gradient of the kernel function, effectively correcting the gradient estimation. These equations represent simplified versions of gradient correction methods used in SPH simulations. The specific form of the equations may vary depending on the details of the method and the properties being calculated. Additionally, there are other more complex and sophisticated gradient correction techniques available in the literature.

3.10. SPH formulation for discontinuity

The Smoothed Particle Hydrodynamics (SPH) method is known for its ability to handle discontinuities in fluid simulations. When a discontinuity, such as a shock wave or material interface, is present in a fluid, the standard SPH formulation may produce spurious numerical artifacts. Several techniques have been developed to address this issue (Gram & Silfwerbrand, 2011; Onyelowe et al., 2022a; Wang et al., 2016). I'll provide a brief overview of two commonly used formulations: the Riemann Solver-based approach and the artificial viscosity method. (i) Riemann Solver-based approach: The Riemann Solver-based approach aims to resolve the discontinuities by explicitly solving the Riemann problem at the fluid interface. The Riemann problem considers the interaction between two states of a fluid separated by a discontinuity. In this approach, the SPH particles on either side of the interface are identified, and the Riemann problem is solved to determine the interfacial properties, such as pressure, velocity, and density (Gram & Silfwerbrand, 2011; Onyelowe et al., 2022a; Wang et al., 2016). Once the interfacial properties are obtained, they are used to modify the standard SPH equations. The modified equations include an additional term, which is derived from the Riemann solution, to account for the discontinuity. This term can be added to the momentum equation, energy equation, or other relevant equations depending on the specific problem being simulated. (ii) Artificial viscosity method: The artificial viscosity method is another commonly used approach to handle discontinuities in SPH simulations. It introduces an artificial viscosity term to the standard SPH equations to mimic the dissipative effects present in real fluids. The artificial viscosity helps to smooth out the discontinuities and prevent spurious oscillations. The artificial viscosity term is added to the momentum equation and is proportional to the velocity difference between neighboring particles. It acts as a shock-capturing mechanism, dissipating energy and smoothing out the flow near the discontinuity. The strength of the artificial viscosity is typically controlled by a viscosity parameter, which needs to be carefully calibrated to balance accuracy and stability. It's worth mentioning that there are other techniques and modifications available to handle discontinuities in SPH simulations, such as the use of limiters, shock capturing schemes, or more advanced formulations like Godunov SPH or Lagrangian Conservative Remap (LCR) SPH. The choice of method depends on the specific problem being simulated and the desired level of accuracy and computational efficiency. When simulating fluid dynamics with the Smoothed Particle Hydrodynamics (SPH) method, handling discontinuities requires modifications to the standard SPH equations (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). Two commonly used formulations for discontinuity treatment in SPH simulations are the Godunov SPH (GSPH) method and the Riemann solver-based approach. I'll provide a brief overview of these formulations and the equations involved. (i) Godunov SPH (GSPH) method: The GSPH method is based on the Godunov approach, which is widely used for capturing discontinuities in computational fluid dynamics. In GSPH, the fluid quantities are treated as piecewise constant within each SPH kernel support region. The discontinuity across the interface between two particles is determined by solving a Riemann problem. The GSPH formulation modifies the momentum and energy equations. The corrected equations for the momentum and energy conservation are as follows:

1. Momentum equation: (20) $${\mathit{\rho }}_{\mathit{i}}\left({\mathit{d}\mathit{v}}_{\mathit{i}}/\mathit{d}\mathit{t}\right)=-{\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\mathit{\rho }}_{\mathit{j}}\right)\left({\mathit{P}}_{\mathit{i}}+{\mathit{P}}_{\mathit{j}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}+{\mathit{\rho }}_{\mathit{i}}\mathit{g}$$(20)

• Energy equation: (21) $${\mathit{\rho }}_{\mathit{i}}\left({\mathit{d}\mathit{e}}_{\mathit{i}}/\mathit{d}\mathit{t}\right)=-{\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\mathit{\rho }}_{\mathit{j}}\right)\left({\mathit{P}}_{\mathit{i}}+{\mathit{P}}_{\mathit{j}}\right){\mathit{v}}_{\text{ij}}{}_{\nabla \mathit{i}}{\mathit{W}}_{\text{ij}}$$(21)

In these equations, ρ_i and ρ_j represent the densities of the interacting particles i and j, respectively. P_i and P_j are the pressures, v_i and v_j are the velocities, and m_j is the mass of particle j. W_ij is the kernel function, and ∇_i represents the gradient with respect to particle i. g denotes any external forces acting on the fluid. (ii) Riemann solver-based approach: The Riemann solver-based approach aims to explicitly solve the Riemann problem across the fluid interface. This approach identifies the particles on either side of the interface and determines the interfacial properties, such as pressure, velocity, and density, by solving the Riemann problem (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). The modified equations for the momentum and energy conservation incorporating the Riemann solver-based approach are as follows:

1. Momentum equation: (22) $${\mathit{\rho }}_{\mathit{i}}\left({\mathit{d}\mathit{v}}_{\mathit{i}}/\mathit{d}\mathit{t}\right)=-{\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\mathit{\rho }}_{\mathit{j}}\right)\left({\mathit{P}}_{\mathit{i}}+{\mathit{P}}_{\mathit{j}}\right){\nabla }_{\mathit{i}}{\mathit{W}}_{\text{ij}}+{\mathit{\rho }}_{\mathit{i}}\mathit{g}+{\mathit{F}}_{\mathit{i}}$$(22)

• Energy equation: (23) $${\mathit{\rho }}_{\mathit{i}}\left({\mathit{d}\mathit{e}}_{\mathit{i}}/\mathit{d}\mathit{t}\right)=-{\sum }_{\mathit{j}}\left({\mathit{m}}_{\mathit{j}}/{\mathit{\rho }}_{\mathit{j}}\right)\left({\mathit{P}}_{\mathit{i}}+{\mathit{P}}_{\mathit{j}}\right){\mathit{v}}_{\text{ij}}{}_{\nabla \mathit{i}}{\mathit{W}}_{\text{ij}}+{\mathit{\rho }}_{\mathit{i}}{\mathit{v}}_{\mathit{i}}{\mathit{F}}_{\mathit{i}}$$(23)

Here, F_i represents the interfacial force terms obtained from the Riemann solution. These terms account for the discontinuity and are applied to the fluid motion and energy evolution equations. It's important to note that these equations provide a simplified overview of the GSPH and Riemann solver-based formulations. The specific implementation and details may vary depending on the specific problem being simulated and the desired accuracy and stability of the simulation.

3.11. Formulation of coupled ANSYS-SPH to model self-compacting concrete flow time

To model the flow time of self-compacting concrete (SCC) using a coupled ANSYS-SPH (Smoothed Particle Hydrodynamics) approach, you can follow the formulation described below. This approach combines the capabilities of ANSYS for structural analysis with the particle-based SPH method for fluid flow simulation (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). (i) Geometry and Meshing: (a) Create a 3D model of the structure or element where SCC flow is being analyzed using ANSYS. (b) Generate a mesh for the structural domain using ANSYS meshing capabilities. Ensure that the mesh is appropriate for capturing the behavior of the solid material. (ii) Material Properties: Define the material properties for the solid phase of SCC using ANSYS. This includes parameters such as density, Young's modulus, Poisson's ratio, and any other relevant material properties. (iii) SPH Particles: (a) Generate a set of SPH particles to represent the fluid phase of SCC. These particles will simulate the flow behavior of SCC. (b) Distribute the particles within the geometry of the ANSYS model. The particle distribution should cover the entire fluid domain. (iv) Coupling: (a) Establish the coupling between ANSYS and the SPH solver. (b) Exchange information between ANSYS and the SPH solver at each time step. This includes transferring forces and displacements between the solid and fluid phases. (v) Fluid Flow Simulation: (a) Apply appropriate boundary conditions to the SPH particles to represent the inlet and outlet of the SCC flow. (b) Simulate the fluid flow using the SPH solver, considering the rheological properties of SCC, such as viscosity and yield stress. (c) Incorporate additional physics, if required, such as turbulence or particle interaction models, to accurately represent the SCC flow behavior. (vi) Structural Analysis: (a) Employ ANSYS to analyze the structural response of the solid phase of SCC. (b) Apply appropriate boundary conditions and loads to the ANSYS model based on the fluid flow results obtained from the SPH simulation. (c) Perform the structural analysis to determine the response of the solid material to the SCC flow. (vii) Iterative Process: (a) Iterate between the SPH simulation and the ANSYS structural analysis until convergence is achieved. (b) Update the fluid flow properties based on the structural response, and vice versa, until a consistent solution is obtained. (viii) Post-processing: (a) Analyze and visualize the results obtained from the coupled ANSYS-SPH simulation. (b) Interpret and present the flow time and other relevant information, such as velocity profiles, pressure distributions, and structural deformations (Hosseinpoor et al., 2017; Sigalotti et al., 2021; Wang et al., 2016). It's important to note that the implementation of a coupled ANSYS-SPH approach for SCC flow time modeling may require custom code development or the use of specialized software that supports both ANSYS and SPH simulations. Additionally, the specific details of the formulation may vary depending on the software and tools being used.

4. Results and discussion

4.1. Geometry and model boundary considerations

4.1.1. General considerations

Figure 2 shows the geometry of the modeled Orimet apparatus. Number of nonhomogeneous particles (coarse and fine) considered in each test setup = infinite, Density/Concrete unit weight = 2400 kg/m³, w/c ratio = 0.43, w/b ratio = 0.43, Flow pattern = under gravity, Fluid property = Bingham non-Newtonian (varying viscosity) incompressible fluid; Model Cases: Case 1: 0% coarse particles and 100% fine particles, Case 2: 60% coarse particles and 40% fine particles, Case 3: 55% coarse particles and 45% fine particles, Case 4: 50% coarse particles and 50% fine particles, Case 5: 45% coarse particles and 55% fine particles, Case 6: 40% coarse particles and 60% fine particles and Case 7: 100% coarse particles and 0% fine particles. The maximum size of the coarse aggregates is considered 20 mm and that of the fine aggregates is below 4 mm. The Bingham model properties for Multiphysics (SPH)-ANSYS models’ simulation are; Viscosity = 20 ≤ μ ≤ 100 and the Yield stress = 50 $\le {\tau }_0\le 200,$ standard flow time, t (s) ranges; 0 ≤ t ≤ 6 and Orimet volume is 10 l.

Figure 2. The Orimet test method geometry.

4.2. Modeling and optimization of Orimet flow cases

The Orimet filling ability of self-compacting concrete (SCC), which refers to its ability to flow and fill all the spaces within a formwork or mold without the need for mechanical compaction has been modeled and optimized in this research work considering the mix percentages of the coarse and fine aggregates in with the EFNARC guidelines (EFNARC, 2002). SCC is designed to have a high flowability and viscosity that allows it to easily flow and pass through congested reinforcement, resulting in complete filling of the formwork under the influence of proportionate heterogeneity sampling of the aggregates. Many other considerations have been adopted previously which included the application of statistical methods and the application of recycled aggregate replacement for the normal aggregate (González-Taboada et al., 2017, 2018; Revilla-Cuesta et al., 2023). These were done considering the appropriate research methodologies (Shukla, 2022). However, to analyze the modeled filling ability of self-compacting concrete, several important factors have been considered: (i) Viscosity: The SCC boundary condition for viscosity; 20 ≤ μ ≤ 100 (EFNARC, 2002) to ensure good filling ability was adhered to. The viscosity can be measured using various methods, such as the V-funnel test or the U-box test. Lower viscosity allows the concrete to flow better and fill the formwork (EFNARC, 2002). (ii) Segregation Resistance: SCC should resist segregation, which is the separation of coarse aggregates from the mortar matrix due to excessive flow. Segregation can lead to poor filling and reduced homogeneity. The stability and resistance to segregation can be evaluated through visual inspection and aggregate segregation tests. (iii) Passing Ability: SCC should have the ability to pass through congested reinforcement without blockage or excessive pressure. The ability to flow around and through reinforcement is crucial for filling complex formwork configurations and this factor was also strictly considered. (iv) Rheological Properties: Rheological properties of SCC, such as yield stress, plastic viscosity, and thixotropy, can provide insights into its filling ability in line with the Bingham plastic model. These properties can be measured using rheometers or viscometers. By considering these factors in this model exercise, the filling ability of self-compacting concrete and the optimized mix design to achieve the desired flowability and filling performance for specific applications was established in terms of the aggregate sampling. When analyzing the filling ability of self-compacting concrete (SCC), the minimum Orimet flow time is an essential parameter, which has been considered. The minimum boundary flow time represents the time; 0 ≤ t ≤ 6, it takes for the SCC to completely flow through a specified distance, typically measured in seconds was modeled for in the seven (7) model cases. The first case; 0%C mixed with 100%F flowed out completely in 6 s, second case; 40%C mixed with 60%F completely flowed out in 5 s, third case; 45%C mixed with 55%F completely flowed out in 9 s, fourth case; 50%C mixed with 50%F completely flowed out in 12 s, fifth case; 55%C mixed with 45%F completely flowed out in 11 s, sixth case; 60%C mixed with 40%F completely flowed out in 12 s, and lastly, the 7th case; 100%C mixed with 0%F completely flowed out in 20 s. The model cases are presented in Figures 3–9, respectively. Each model scenario shows the start and end of the flow time. These show an indicator of the SCC's ability to rapidly fill formwork or molds without the need for additional compaction at minimum flow time (OFT). To analyze the model filling ability based on the minimum flow time of SCC, the following steps have been taken: (i) Measurement and Analysis: Start the timing when the SCC starts to flow from the bottom of the Orimet and stop the timing when the flow ceases. Measure the time in seconds and record it as the minimum flow time. (ii) Interpretation: The minimum flow time provides insight into the filling ability of the SCC. A shorter minimum flow time indicates better flowability and faster filling characteristics. It suggests that the SCC can easily flow and fill complex formwork or molds, including congested reinforcement, without the need for additional compaction. (iii) Comparison and Optimization: Compare the obtained minimum flow time with the desired specifications or requirements for the specific application. If the minimum flow time is not within the acceptable range, adjustments can be made to the mix design, such as modifying the proportions of the constituents, especially the aggregate grading followed by water content, or the use of chemical admixtures, to improve the SCC's filling ability. The flow interpretation from the seven (7) model cases and comparison of the model outcomes show that the flow time has been optimized with 40%C mixed with 60%F aggregate sampling in the mix flowing at 5 s and this is supported by the guidelines provided by the EFNARC (EFNARC, 2002) that allows 0–6 s as the standard flow time through the Orimet. It is important to note that the minimum flow time has been considered alongside other relevant parameters and tests, such as slump flow, passing ability, segregation resistance, and rheological properties (stresses), to comprehensively assess the filling ability of SCC in this model. By considering these factors and the optimized mix (40%C + 60%F:5 s), engineers and researchers can optimize the SCC mix design to achieve the desired flowability and filling performance for their specific construction applications. The proportion of coarse aggregates and fine aggregates in self-compacting concrete (SCC) can significantly influence its filling ability and flow time. Generally, SCC contains a higher proportion of fine aggregates compared to conventional concrete mixes. However, the specific effects of a 40% coarse aggregate and 60% fine aggregate mix on the filling ability Orimet flow time of SCC will depend on various factors, such as the properties of the aggregates, the mix design, and the desired flowability. Here are a few potential influences of a 40% coarse aggregate and 60% fine aggregate mix on the filling ability flow time of SCC: (i) Flowability: The higher proportion of fine aggregates can improve the flowability of SCC. Fine aggregates, such as sand, have smaller particle sizes compared to coarse aggregates. This can enhance the lubrication and filling ability of the concrete mixture, allowing it to flow more easily through congested reinforcement and complex formwork. (ii) Viscosity and Segregation Resistance: The increased amount of fine aggregates can influence the viscosity and segregation resistance of SCC. Finer particles tend to increase the paste content and improve the cohesion of the concrete mixture. This can enhance the stability of SCC, reducing the risk of segregation during flow and improving its filling ability. (iii) Packing Density: The packing density of aggregates can affect the flowability of SCC. Coarse aggregates generally have lower packing densities compared to fine aggregates. Therefore, a higher proportion of fine aggregates may result in a higher overall packing density, improving the flowability and filling ability of SCC. (iv) Aggregate Grading: The specific grading of the coarse and fine aggregates will also impact the filling ability flow time of SCC. The grading should be optimized to achieve a well-balanced particle size distribution, ensuring good particle packing and reducing the potential for blockages during flow. Proper grading can contribute to improved filling ability and reduced flow time. (v) Workability Retention: The 40% coarse aggregates and 60% fine aggregates mix can affect the workability retention of SCC. Workability retention refers to the ability of the concrete to maintain its desired flowability over time. The fine aggregates can help in maintaining the required workability for an extended period, allowing the SCC to fill the formwork adequately without significant loss of flowability. It is important to note that the specific effects of a 40% coarse aggregate and 60% fine aggregate mix on filling ability flow time may vary depending on the specific properties of the aggregates, the mix proportions, and the desired flowability requirements (Onyelowe, Kontoni, Onyia, et al., 2023). Conducting laboratory tests and trials using the actual materials and mix design will provide more insights into the filling ability and flow time of SCC in this particular scenario.

Figure 3. 0%C mixed with 100%F case multiphysics model with Orimet flow time (OFT) of 6 s.

Figure 4. 40%C mixed with 60%F case multiphysics model with Orimet flow time (OFT) of 5 s.

Figure 5. 45%C mixed with 55%F case multiphysics model with Orimet flow time (OFT) of 9 s.

Figure 6. 50%C mixed with 50%F case multiphysics model with Orimet flow time (OFT) of 12 s.

Figure 7. 55%C mixed with 45%F case multiphysics model with Orimet flow time (OFT) of 11 s.

Figure 8. 60%C mixed with 40%F case multiphysics model with Orimet flow time (OFT) of 12 s.

Figure 9. 100%C mixed with 0%F case multiphysics model with Orimet flow time (OFT) of 20 s.

4.3. ANSYS-SPH interface model simulation of optimized Orimetflow

The multiphase optimized mix (40%C + 60%F:5 s) was further simulated using the coupled interface of the ANSYS-SPH platform operating with the CFX command at air temperature of 25 °C, which incorporated the studied density of 2400 kg/m³, plastic viscosity boundary, yield stress, and aggregate sampling. This was conducted using the analytical space configuration of the smoothed particle hydrodynamics embedded in the fluid dynamic manipulation of the ANSYS solver. The model simulation operated on total number of nodes = 143,083, total number of elements = 753,292, total number of tetrahedrons = 753,292, and total number of faces = 68,488 and produced Dynamic Viscosity = 1.831E-05 kg m⁻¹ s⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Absorption Coefficient = 0.01 m⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Refractive Index = 1.0 m m⁻¹, Molar Mass = 1 kg kmol⁻¹, Specific Heat Capacity = 8.80E + 02 J kg⁻¹ K⁻¹, Normal Speed = 165 mm s⁻¹, Pressure Profile Blend = 0.05, and Maximum Partition Smoothing Sweeps = 100. Also, the Global Length= 1.9144E-01, Minimum Extent = 1.1800E-01, Maximum Extent = 6.5976E-01, Density = 1.1850E + 00, Velocity = 1.6500E-01, Advection Time = 1.1602E + 00, and Reynolds Number = 2.0443E + 03. To simulate the optimized flow time and filling ability of a Self-Compacting Concrete (SCC) mixture with 40% coarse aggregate and 60% fine aggregate, the coupled interface tool that was specifically designed for concrete mix design and simulation was employed. These tools take into account various factors such as aggregate properties, water-cement ratio, admixtures, and other parameters to predict the flow time and filling ability of the concrete mix. Figures 10–13 show the simulation results. Figure 10 shows the discretization of the Orimet model apparatus, Figure 11 shows the simulated velocity, pressure, total pressure dissipation, turbulence kinetic energy and the velocity streamline, while Figures 11 and 12 present the graphical behavior of these simulated Orimet flow characteristics at 5 s of optimized flow time. The simulation produced wall forces and moment on the wall of the Orimet for the optimized mix containing 40%C + 60%F:5s flow mix as follows; pressure force on wall; −3.0996E-08, −2.0863E-07, and −3.5048E-04 for x-component, y-component, and z-component, respectively, viscous force on wall; −5.5332E-10, −9.2298E-10, and −2.7250E-05 for the x-, y-, and z-components, respectively, pressure moment on wall; −5.0051E-05, 3.1362E-06, and 2.2774E-09 for the x-, y-, and z-components, respectively and viscous moment on wall; −3.8925E-06, 2.4396E-07, and −7.2693E-11 for the x-, y-, and z-components, respectively. Also, the maximum residuals were located at node 110413 for the pressure, node 76766 for the K-TurbKE, and node 110724 for the E-Diss.K.

Figure 10. Discretization of the Orimet apparatus.

Figure 11. Orimet flow velocity and energy properties for the optimized mix.

Figure 12. Graphical representation of the optimized flow energy and velocity properties.

Figure 13. Flow interparticle properties and on the wall.

5. Conclusions

In this research work seven (7) model Cases: Case 1: 0% coarse particles and 100% fine particles, Case 2: 60% coarse particles and 40% fine particles, Case 3: 55% coarse particles and 45% fine particles, Case 4: 50% coarse particles and 50% fine particles, Case 5: 45% coarse particles and 55% fine particles, Case 6: 40% coarse particles and 60% fine particles and Case 7: 100% coarse particles and 0% fine particles have been studied to model the influence of proportionate heterogeneity of aggregates on the Orimet flow time of SCC. The maximum size of the coarse aggregates is considered 20 mm and that of the fine aggregates is below 4 mm and the following can be concluded;

• The Bingham model properties for the Multiphysics (SPH)-ANSYS models’ simulation are; Viscosity = 20 ≤ μ ≤ 100 and the Yield stress = 50 $\le {\tau }_0\le 200,$ standard flow time, t (s) ranges; 0 ≤ t ≤ 6 and Orimet volume is 10 l.

• The minimum boundary flow time represents the time; 0 ≤ t ≤ 6, it takes for the SCC to completely flow through a specified distance, typically measured in seconds was modeled for in the seven (7) model cases. The first case; 0%C mixed with 100%F flowed out completely in 6 s, second case; 40%C mixed with 60%F completely flowed out in 5 s, third case; 45%C mixed with 55%F completely flowed out in 9 s, fourth case; 50%C mixed with 50%F completely flowed out in 12 s, fifth case; 55%C mixed with 45%F completely flowed out in 11 s, sixth case; 60%C mixed with 40%F completely flowed out in 12 s, and lastly, the 7th case; 100%C mixed with 0%F completely flowed out in 20 s.

• The minimum flow time was considered alongside other relevant parameters and tests, such as slump flow, passing ability, segregation resistance, and rheological properties (stresses), to comprehensively assess the filling ability of SCC in this model. By considering these factors and the optimized mix (40%C + 60%F:5 s), engineers and researchers can optimize the SCC mix design to achieve the desired flowability and filling performance for their specific construction applications.

• The multiphase optimized mix (40%C + 60%F:5 s) was further simulated using the coupled interface of the ANSYS-SPH platform operating with the CFX command at air temperature of 25 °C, which incorporated the studied density of 2400 kg/m³, plastic viscosity boundary, yield stress, and aggregate sampling. The model simulation operated on total number of nodes = 143,083, total number of elements = 753,292, total number of tetrahedrons = 753,292, and total number of faces = 68,488 and produced Dynamic Viscosity = 1.831E-05 kg m⁻¹ s⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Absorption Coefficient = 0.01 m⁻¹, Thermal Conductivity = 2.61E-02 W m⁻¹ K⁻¹, Refractive Index = 1.0 m m⁻¹, Molar Mass = 1 kg kmol⁻¹, Specific Heat Capacity = 8.80E + 02 J kg⁻¹ K⁻¹, Normal Speed = 165 mm s⁻¹, Pressure Profile Blend = 0.05, and Maximum Partition Smoothing Sweeps = 100. Also, the Global Length = 1.9144E-01, Minimum Extent = 1.1800E-01, Maximum Extent = 6.5976E-01, Density = 1.1850E + 00, Velocity = 1.6500E-01, Advection Time = 1.1602E + 00, and Reynolds Number = 2.0443E + 03.

• Also, the simulation produced wall forces and moment on the wall of the Orimet for the optimized mix containing 40%C + 60%F:5 s flow mix as follows; pressure force on wall; −3.0996E-08, −2.0863E-07, and −3.5048E-04 for x-component, y-component and z-component, respectively, viscous force on wall; −5.5332E-10, −9.2298E-10, and −2.7250E-05 for the x-, y-, and z-components, respectively, pressure moment on wall; −5.0051E-05, 3.1362E-06, and 2.2774E-09 for the x-, y-, and z-components, respectively and viscous moment on wall; −3.8925E-06, 2.4396E-07, and −7.2693E-11 for the x-, y-, and z-components, respectively. Also, the maximum residuals were located at node 110,413 for the pressure, node 76,766 for the K-TurbKE, and node 110,724 for the E-Diss.K.

• It is suggested that this model work be extended to the L-box, V-funnel, J-ring, etc. flow apparatus as part of a future research focus.

Ethical approval

Not applicable.

Consent for publication

The authors give their consent for the publication of this research paper.

Author contributions

KCO conceptualized, KCO and D-PNK wrote the main manuscript text and both authors reviewed the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data supporting this research work is available on reasonable request from the corresponding author.

Additional information

Notes on contributors

Kennedy C. Onyelowe

Kennedy C. Onyelowe is an Associate Professor of Civil Engineering specializing in AI, sustainable construction materials, concrete and geotechnics at the Michael Okpara University of Agriculture, Umudike, Nigeria, and Kampala International University, Kampala, Uganda and a research fellow for a doctoral diploma at the University of the Peloponnese, Patras, Greece.

Denise-Penelope N. Kontoni

Denise-Penelope N. Kontoni is an Associate Professor of Civil Engineering specializing in computational structural dynamics, AI, structures and geotechnics at the University of the Peloponnese, Patras, Greece.