Numerical Simulation of the Stability of Low Viscosity Ratio Viscoelastic Lid-Driven Cavity Flow Based on the Log-Conformation Representation (LCR) Algorithm

Abstract

Log-Conformation Representation (LCR) method effectively enhances the stability of viscoelastic fluid flow driven by a cavity at high Wi numbers. However, its stability is relatively poor under low viscosity ratio conditions. In this study, three momentum equation stabilization algorithms (Both-Sides-Diffusion, Discrete Elastic Viscous Split Stress-Vorticity, and velocity–stress coupling) were tested and compared in OpenFOAM to assess their stabilizing effects on the LCR method under low viscosity ratio conditions. The evaluation was based on changes in average kinetic energy and the maximum critical time step. The results indicate that the different momentum equation stabilization algorithms improve the numerical oscillations observed in the numerical simulation of low viscosity ratio cavity-driven flow to varying extents. This enables a reduction in the viscosity ratio that can be stably simulated by 0.03 to 0.15. Furthermore, these cases using the momentum equation stabilization algorithms require time steps that are 33% to 100% shorter than those of the original cases. This demonstrates the promoting effect of the additional diffusion term in the momentum equation on stability under low viscosity ratio conditions. The combination of LCR and velocity–stress coupling was used to analyze the impact of viscosity ratios on velocity, logarithmic conformation tensor, and average kinetic energy. As the viscosity ratio decreases, the contribution of fluid elasticity increases, resulting in more pronounced variations in velocity and stress. However, the viscosity ratio has little effect on the stress boundary layer at the top cover and corners. Under conditions with the same Wi number, the average kinetic energy decreases as the viscosity ratio decreases until stability is achieved.

1. Introduction

Viscoelastic flow has a wide range of applications in natural sciences and engineering practices. However, numerical simulations of viscoelastic fluids often encounter convergence issues when the Weissenberg (Wi) number surpasses a critical threshold, resulting in a substantial rise in computational effort and eventual breakdown. This phenomenon is commonly known as the “High Weissenberg Number Problem” (HWNP) [1].

In order to address the HWNP, researchers have proposed various improvement methods, including the Discontinuous Galerkin (DC) [2], Streamline-Upwind Petrov Galerkin (SUPG) [3], and SRCR methods [4], among others. These methods effectively enhance the stability of computations and improve the HWNP, but they still cannot completely solve the problem. Among the prevailing computational approaches, the Log-Conformation Representation (LCR) method, proposed by Fattal and Kupferman [5,6], has garnered attention. This method reformulates the conformation tensor in the constitutive equation of viscoelastic fluids using a logarithmic formulation, linearizing the exponential stress distribution, and preventing the exponential growth of the conformation tensor. Consequently, it significantly improves numerical stability at high Wi numbers while preserving the positive definiteness and symmetry of the conformation tensor during computations. The LCR method exhibits excellent performance in terms of computational stability and accuracy under high Wi conditions [7,8,9,10,11].

Lid-driven cavity flow, as a commonly encountered flow in industrial applications, has received significant attention. Extensive research has been conducted on cavity-driven flow of Newtonian fluids from various perspectives [12,13,14,15]. Similarly, substantial research has been carried out on cavity-driven flow of viscoelastic fluids, exploring different aspects such as constitutive equations [16], cavity shapes [17], heat exchange [18], and electroconvection [19]. On the other hand, the flow-driven problem of Oldroyd-B fluid in a square cavity is a classic benchmark. Although its geometric shape is simple, the fluid driven by the top lid inside the cavity is affected by complex flow characteristics, including shear deformation and elongation deformation [20]. In addition, the simulation of the flow driven by the square cavity also encounters the problem of high Weissenberg number due to the stress singularity near the top lid and the corners of the side walls. Therefore, it is considered a very rigorous numerical method testing problem. Fattal and Kupferman [6] proposed the LCR method and simulated the flow driven by the square cavity of Oldroyd-B fluid as a benchmark case, improving the maximum Wi number that can be achieved by numerical simulation of the flow driven by the square cavity, verifying the stability of the LCR method at high Wi numbers, and providing reference data. After that, Comminal et al. [20] used a new stream function/log-conformation formulation and simulated the two-dimensional flow of Oldroyd-B fluid in the flow driven by the square cavity at a wide range of Wi numbers. Pan et al. [21] introduced the logarithmic conformation technique into the finite element method using the operator splitting method to simulate the time-dependent flow of Oldroyd-B fluid in the flow driven by the square cavity. Zhou et al. [22] used unstructured triangular grids to simulate the flow driven by the square cavity of Oldroyd-B fluid and studied the effect of flow elasticity. Viscoelastic lid-driven cavity flow has been extensively studied in engineering applications and numerical validation. However, there is limited research specifically focusing on low viscosity ratio conditions.

The LCR has demonstrated effective improvements in addressing the HWNP in driven cavity flow. However, the stability of simulations and flow characteristics under low viscosity ratio conditions have received limited attention. Chen et al. [23] identified certain limitations of the LCR method in their numerical investigation, particularly in achieving stability under low viscosity ratio conditions. Since the LCR method essentially involves a logarithmic transformation of the constitutive equation of viscoelastic fluids without modifying the momentum equation, the ellipticity of the momentum equation weakens under low viscosity ratio conditions, which hinders the stability of numerical iterations and leads to poor performance of the LCR method for such problems. To address numerical simulations under low viscosity ratio conditions, an additional diffusion term is added to both sides of the momentum equation to maximize the elliptic operator, thereby enhancing the ellipticity of the momentum equation. This modification improves the convergence of iterative solving algorithms, making it an effective class of algorithms [23,24,25,26]. These algorithms can be easily implemented by transforming the momentum equation, eliminating the need for complex algorithm derivation and implementation. Examples of such algorithms include Both-Sides-Diffusion (BSD) [27], discrete elastic viscous split stress-vorticity (DEVSS-ω) [28], and velocity–stress coupling [29].

The paper is structured as follows: Section 2 introduces the fundamental equations of the Oldroyd-B fluid, accompanied by a concise overview of the LCR method and the stable momentum equation approach. In Section 3, we provide a brief elucidation of the cavity-driven flow problem and undertake a comparative analysis of the stability of cavity-driven flow at low viscosity ratios under distinct algorithmic influences. Moreover, building upon this analysis, we investigate the flow characteristics of cavity-driven flow across different viscosity ratios. Conclusions are finally presented in Section 4.

2. Governing Equations and Methods

2.1. Governing Equations

The governing equations of isothermal, incompressible viscoelastic fluids are given by the following Equations (1) and (2):

$$\nabla ·\mathit{u}=0$$

(1)

$$\rho (\frac{\partial \mathit{u}}{\partial t}+\mathit{u}·\nabla \mathit{u})=-\nabla p+{\eta }_s{\nabla }^2\mathit{u}+\nabla ·{\mathit{\tau }}_p$$

(2)

where u is the velocity vector, ρ is the fluid density, t is the time, p is the pressure, τ_p is the elastic stress, and η_s is the solvent viscosity. The elastic stress in the Oldroyd-B model is described by the following hyperbolic partial differential equation:

$$\frac{\partial {\mathit{\tau }}_p}{\partial t}+\mathit{u}·\nabla {\mathit{\tau }}_p={\mathit{\tau }}_p·\nabla \mathit{u}+{(\nabla \mathit{u})}^{\mathbf{T}}·{\mathit{\tau }}_p+\frac{1}{\lambda }\{2{\eta }_p[\nabla \mathit{u}+{(\nabla \mathit{u})}^{\mathrm{T}}]-{\tau }_p\}$$

(3)

with λ the relaxation time, η_p is the polymer viscosity. The viscosity ratio is $\beta ={\eta }_s/({\eta }_s+{\eta }_p)$.

2.2. Log-Conformation Representation

The relationship between the conformation tensor (C) and the elastic stress tensor τ_p in the Oldroyd-B model is given by the following equation:

$${\tau }_p=\frac{{\eta }_p}{\lambda }(\mathit{C}-\mathbf{I})$$

(4)

where I represents the identity tensor. Combining the constitutive Equation (3) of the Oldroyd-B model, the equation for the conformation tensor in the Oldroyd-B model can be obtained as follows:

$$\frac{D\mathit{C}}{Dt}=\frac{\partial \mathit{C}}{\partial t}+\mathit{u}·\nabla \mathit{C}=\mathit{C}·\nabla \mathit{u}+{(\nabla \mathit{u})}^{\mathbf{T}}·\mathit{C}+\frac{1}{\lambda }(\mathbf{I}-\mathit{C})$$

(5)

where $\mathit{C}=\mathit{R}·\mathit{\Lambda }·{\mathit{R}}^{\mathrm{T}}$. Defining $\frac{D}{Dt}=\frac{\partial }{\partial t}+(\mathit{u}·\nabla )$, $\mathit{L}=\nabla \mathit{u}$. Substituting it into Equation (5), we obtain:

$$\frac{D\mathit{C}}{Dt}=\mathit{C}·\mathit{L}+{\mathit{L}}^{\mathbf{T}}·\mathit{C}+\frac{1}{\lambda }(\mathbf{I}-\mathit{C})$$

(6)

From $\mathit{\Theta }=log\mathit{C}=\mathit{R}·log\mathit{\Lambda }·{\mathit{R}}^{\mathbf{T}}$ to Equation (6) we can obtain

$$\tilde{\mathit{S}}·\mathit{\Lambda }+\frac{D\mathit{\Lambda }}{Dt}+\mathit{\Lambda }·{\tilde{\mathit{S}}}^{\mathbf{T}}=\tilde{\mathit{L}}·\mathit{\Lambda }+\mathit{\Lambda }·{\tilde{\mathit{L}}}^{\mathbf{T}}+\frac{1}{\lambda }\mathbf{I}-\frac{1}{\lambda }\mathit{\Lambda }$$

(7)

$$\frac{D\mathit{\Theta }}{Dt}=\mathit{R}·(\tilde{\mathit{S}}·log\mathit{\Lambda }+\frac{D\mathit{\Lambda }}{Dt}·{\mathit{\Lambda }}^{-1}+log\mathit{\Lambda }·{\tilde{\mathit{S}}}^{\mathrm{T}})·{\mathit{R}}^{\mathrm{T}}$$

(8)

where: $\tilde{\mathit{S}}={\mathit{R}}^{\mathrm{T}}·\frac{D\mathit{R}}{Dt}$, $\tilde{\mathit{L}}={\mathit{R}}^{\mathrm{T}}·\mathit{L}·\mathit{R}$.

By simultaneously combining Equations (7) and (8), we can obtain the equation for the logarithmic conformation tensor in the Oldroyd-B model:

$$\frac{D\mathit{\Theta }}{Dt}=\mathit{S}·\mathit{\Theta }-\mathit{\Theta }·\mathit{S}+2\mathit{{\rm B}}+\frac{1}{\lambda }(e^{-\mathit{\Theta }}-\mathbf{I})$$

(9)

where $\mathit{B}=\mathit{R}·\tilde{\mathit{B}}·{\mathit{R}}^{\mathrm{T}}$ (${\tilde{B}}_{ii}={\tilde{L}}_{ii}$, ${\tilde{B}}_{ij}=0$) and $\mathit{S}=\mathit{R}·\tilde{\mathit{S}}·{\mathit{R}}^{\mathrm{T}}$.

2.3. Numerical Stabilization Algorithms

2.3.1. Both-Sides-Diffusion [27] (BSD)

As η_s → 0, the elliptic diffusion term on the right-hand side of Equation (2) reduces to zero in the momentum equation. Due to the absence of a clear diffusion term, these equations are difficult to converge. To overcome this limitation, an additional diffusion term is added on both sides of the momentum equation. Therefore, the momentum Equation (2) is modified as follows:

$$\rho (\frac{\partial \mathit{u}}{\partial t}+\mathit{u}·\nabla \mathit{u})-({\eta }_s+{\eta }_p){\nabla }^2\mathit{u}=-\nabla p+\nabla ·{\mathit{\tau }}_p-{\eta }_p{\nabla }^2\mathit{u}$$

(10)

This algorithm of adding diffusion terms is commonly referred to as the BSD (Both-Sides-Diffusion) method. In the above equations, all terms on the left-hand side of the equations are discretized implicitly, while the remaining terms on the right-hand side are discretized explicitly as a source term. In the case of numerical convergence, these additional terms cancel each other out.

2.3.2. Discrete Elastic Viscous Split Stress Vorticity [28] (DEVSS-ω)

In the momentum equation, in order to make the solution of the governing equation more stable, the DEVSS-ω (Discrete Elastic Viscous Stress Splitting with Vorticity) method is employed, which utilizes the following identity in the momentum equation:

$${\nabla }^2\mathit{u}=-\nabla \times \mathit{\omega }+\nabla (\nabla ·\mathit{u})$$

(11)

where $\mathit{\omega }=\nabla \times \mathit{u}$. Adding this to Equation (2), we obtain

$$\rho (\frac{\partial \mathit{u}}{\partial t}+\mathit{u}·\nabla \mathit{u})-({\eta }_s+\epsilon ){\nabla }^2\mathit{u}=-\nabla p+\nabla ·{\mathit{\tau }}_p+\epsilon \nabla \times \mathit{\omega }-\epsilon \nabla (\nabla ·\mathit{u})$$

(12)

where ε is the diffusion coefficient, and in this paper, ε = η_p. In the discretization practice, the value of the vorticity at a node is obtained using its definition, and its value is estimated using linear interpolation on the grid when computing the source term of the resulting equation. The left-hand side term of Equation (12) is treated implicitly, while the right-hand side term is discretized explicitly. The additional diffusion term discretized implicitly enhances the ellipticity of the equation, thus aiding in the stability of the numerical iteration.

2.3.3. Velocity–Stress Coupling [29] (Coupling)

There is a stress–velocity decoupling issue in numerical simulations of viscoelastic fluids. The velocity at the grid point loses the influence of forces in its immediate neighborhood. This situation often occurs during the interpolation from a grid-centered to a face-centered field. This problem can arise when computing the divergence of the elastic stress tensor, denoted as $\nabla ·{\mathit{\tau }}_p$ in the momentum equation, which requires the values of the elastic stress on the grid surface. In the finite volume method, the discretization of $\nabla ·{\mathit{\tau }}_p$ is given by:

$${\displaystyle \underset{V}{\int }\nabla ·{\mathit{\tau }}_p}\mathrm{d}V={\displaystyle \sum _f{\overrightarrow{\mathrm{S}}}_f}·{\mathit{\tau }}_{p,f}$$

(13)

where ${\overrightarrow{\mathrm{S}}}_f$ is the product of the grid surface area and the normal vector, and ${\mathit{\tau }}_{p,f}$ is the value of the elastic stress on the grid surface, which is obtained by interpolating from the values at neighboring grid centers. The stress–velocity coupling in the momentum equation is a novel algorithm that decomposes the value of elastic stress on the grid surface, promoting the coupling between stress and velocity. The value of elastic stress at the center of the grid surface is decomposed into:

$${\mathit{\tau }}_{p,f}={\overline{\mathit{\tau }}}_{p,f}+{\eta }_p[(\nabla \mathit{u}{|}_f+{(\nabla \mathit{u})}^{\mathrm{T}}{|}_f)-(\overline{\nabla \mathit{u}}{|}_f+{(\overline{\nabla \mathit{u}})}^{\mathrm{T}}{|}_f)]$$

(14)

where $\overline{\nabla \mathit{u}}$ is linearly interpolated from the values at the grid centers, and the remaining velocity gradients are directly computed from the velocity values centered on the grid cells spanning the faces. Substituting Equation (14) into the momentum equation contributes to the stress–velocity coupling. It also enhances the ellipticity of the momentum equation, thereby increasing stability. The resulting new momentum equation is:

$$\rho (\frac{\partial \mathit{u}}{\partial t}+\mathit{u}·\nabla \mathit{u})-({\eta }_s+{\eta }_p){\nabla }^2\mathit{u}=-\nabla p+\nabla ·{\overline{\mathit{\tau }}}_p-\nabla ·\overline{({\eta }_p\nabla \mathit{u})}$$

(15)

where $\nabla ·\overline{({\eta }_p\nabla \mathit{u})}$ is a “special second-order derivative” (different from the Laplacian operator in OpenFOAM), defined as the divergence of the velocity gradient. The velocity gradient at the face is obtained by linearly interpolating the evaluated velocity gradient at the grid center.

2.4. Numerical Method

In this paper, numerical simulations were carried with the open-source CFD toolbox OpenFOAM-9. The convective term is discretized using the CUBISTA scheme [30]. For viscoelastic fluid flow, a delayed correction method is also employed. The diffusion term in the momentum equation is discretized using a central differencing scheme, while the source term is discretized using second-order Gaussian discretization. The velocity gradients are calculated using Gaussian linear interpolation. For the temporal discretization, a Crank–Nicolson scheme is used since the case studies considered in this paper are focused on steady-state solutions. The transient term used in the momentum equation is only employed for time advancement purposes and vanishes when the steady-state condition is reached. The pressure Poisson equation is solved using the preconditioned conjugate gradient method with a diagonal-based incomplete Cholesky preconditioner (DIC). The velocity and log-conformation tensor equations are solved using the preconditioned biconjugate gradient method (PBiCG) with a diagonal-based incomplete LU preconditioner. The coupled calculation is performed using the SIMPLEC algorithm [31].

3. Results

3.1. Problem Description

In this case study, the suitability of different stability algorithms was evaluated for simulating lid-driven flow in a cavity. The geometric structure and mesh diagram is shown in Figure 1, which consists of a two-dimensional rectangular cavity with H = 1 m. The flow inside the cavity is driven by tangential motion of the lid, while the other boundaries are subjected to the no-slip condition. To compare the simulation results with the reference, the following regularized velocity profile for the lid motion was adopted, where the lid translation velocity distribution is given by the following equation:

$$W(x,t)=8[1+\tanh 8(t-0.5)]U{x}^2(1-x^2)$$

where U = 1 m/s. The setting of the lid translation velocity is to prevent the occurrence of infinite stress gradients at the corners of the cavity, and the regularized velocity distribution can be physically interpreted as the phenomenon of fluid sliding on the moving lid. Additionally, the hyperbolic tangent function makes the acceleration change more gradual, eliminating the influence of sudden flow field changes at the initial time. When t > 1 s, the lid velocity reaches a steady-state value [20]. The simulated conditions in this paper are limited to creeping flow conditions ($Re=\rho UH/{\eta }_0\to 0$), where the convective term of the momentum equation is neglected. $Wi=\lambda U/H$ is defined based on the center velocity U and width H of the lid translation, while the Weissenberg number (Wi) varies by changing the value of the relaxation time (λ). The viscosity ratio (β) is varied by changing the values of η_s and η_p while keeping η_s + η_p constant.

The kinetic energy can serve as a reference benchmark for comparing the simulation results with the reference [6]. It is defined as follows:

$$E_k=\frac{\frac{1}{2}{\displaystyle {\int }_{\mathrm{\Omega }}\left|\mathit{u}·\mathit{u}\right|}d\mathrm{\Omega }}{{\displaystyle {\int }_{\mathrm{\Omega }}d\mathrm{\Omega }}}$$

3.2. Grid Independence Verification

In order to obtain accurate results and save computational costs in the simulation, grid convergence of the numerical solution was investigated using four different grids. Four types of uniform grids were considered. These grids are divided equally. The details are shown in the following (

Table 1. Mesh information for lid-driven cavity.

Mesh	Number of Control Volumes	Δx/H = Δy/H
M1	9216	1.04 × 10−2
M2	16,384	7.8 × 10−3
M3	36,864	5.21 × 10−3
M4	65,536	3.91 × 10−3

):

The computational conditions for grid verification are as follows: Wi = 2, 3 and β = 0.5. In this paper, the velocity component v/U at y/H = 0.75, the logarithmic conformation tensor component θ_xy, and the kinetic energy E_k were selected as variables to investigate the grid independence of the numerical solution. The time step was set to s until the simulation results reached a steady state. This section employed the LCR method for simulation, and the simulation results are shown in the figure.

Figure 2 demonstrates that as the number of grid points increases, v/U, θ_xy, and E_k gradually approach constant values. The numerical results obtained on the M3 grid are very close to those obtained on the M4 grid, and they are also consistent with the results reported in the reference [6]. To strike a balance between computational cost and accuracy, the M3 grid will be used for subsequent numerical simulations.

3.3. Accuracy Verification

To validate the effectiveness of combining different stability algorithms with the LCR method, this study conducted a comparison based on the case presented as an example of lid-driven cavity for Oldroyd-B fluid in reference [6]. This example serves as a classic benchmark case widely used for validating numerical algorithms. The computational conditions for this verification were Wi = 2, 3 and β = 0.5. The boundary conditions are the same as those mentioned earlier in the text. For larger Wi values, a longer simulation time is required to obtain a stable flow field. Figure 3 shows the simulation results on an M3 grid at t = 40 s for Wi = 2 using LCR, BSD + LCR, coupling + LCR, and DEVSS-ω + LCR. The velocity and stress variations in the LCR method combined with different algorithms exhibit smooth curves without any significant variations. The plots of kinetic energy variations for different algorithms demonstrate similar trends. The obtained stable numerical solution is in good agreement with the results reported in reference [6]. As shown in Figure 4, the simulation results for Wi = 3 do not exhibit quasi-periodic oscillations as reported in reference [6]. Instead, they reach a steady state as time progresses. The overall trends of velocity and the logarithmic conformation tensor along the x and y directions are similar to those reported in reference [6]. However, there are differences in the regions near the wall where the stress variations are more pronounced. Nevertheless, the overall trends remain similar.

3.4. Analysis of the Simulation Results for Low Viscosity Ratio Lid-Driven Flow in a Cavity

3.4.1. Stability Analysis of the Simulation of Low Viscosity Ratio Lid-Driven Flow in a Cavity

In order to compare the stabilizing effects of different stability algorithms on the LCR method for low viscosity ratio lid-driven flow in a cavity, this study employs LCR, LCR + BSD, LCR + coupling, and LCR + DEVSS-ω. The simulations are conducted under Wi conditions of 1, 2, and 3.

By varying the parameter β, the study aims to determine the minimum viscosity ratio at which stability can be achieved in the simulations using different stability algorithms.

Table 2. Critical viscosity ratio β_cri of lid-driven cavity by using different steady-state algorithms under different Wi numbers.

	LCR	LCR + BSD	LCR + Coupling	LCR + DEVSS-ω
Wi = 1	0.09	0.06	0	0.03
Wi = 2	0.28	0.22	0.2	0.2
Wi = 3	0.43	0.32	0.25	0.25

provides an overview of the minimum viscosity ratios obtained with the different stability algorithms. It is observed that at lower Wi values, the simulations can achieve lower viscosity ratios. Furthermore, the utilization of stability algorithms enables the attainment of stable lid-driven flow even at lower viscosity ratios.

It should be noted that simulations can still be conducted beyond this critical viscosity ratio. However, it should be noted that as the viscosity ratio increases, periodic oscillations occur in regions of high stress, particularly at the lid and corners of the cavity. In order to characterize this phenomenon and demonstrate the stabilizing effect of different stability algorithms on the LCR method under low viscosity ratio conditions, this study uses the average kinetic energy as a representation. The specific results are shown in Figure 5, Figure 6 and Figure 7.

It is important to acknowledge that simulations can be conducted beyond the critical viscosity ratio mentioned earlier. However, it should be noted that as the viscosity ratio increases, periodic oscillations occur in regions of high stress, particularly at the lid and corners of the cavity. These oscillations become more pronounced with higher viscosity ratios. To characterize this phenomenon and illustrate the stabilizing effect of different stability algorithms on the LCR method under low viscosity ratio conditions, this study utilizes the average kinetic energy as a representation. The specific results of this analysis are presented in Figure 5, Figure 6 and Figure 7.

At Wi = 1 and β = 0.1, all of the methods employed in the simulations reach a steady state and display similar trends over time. However, there are differences in the final stable values among the simulations utilizing different algorithms. From the simulation results at β = 0.05, it becomes evident that the LCR method alone exhibits a certain level of periodic oscillations. Conversely, the simulations incorporating momentum equation stability algorithms maintain stability. This observation indicates the limitations of the LCR method under low viscosity ratio conditions and highlights the improved stability achieved by employing momentum equation stability algorithms. As β is further decreased, the periodic oscillations in the simulations utilizing the original LCR method become more severe. Additionally, oscillations are also observed in the simulations using LCR + DEVSS-ω, and LCR + BSD methods. However, the simulations using the LCR + coupling method remain stable.

The simulation results at Wi = 2, as shown in Figure 6, indicate that oscillations occur in all algorithms at β = 0.1. However, the oscillations are more pronounced in the LCR + DEVSS-ω and LCR + coupling simulations, deviating from the observed trends observed in the Wi = 1 simulations. At β = 0.2, the simulations employing LCR + BSD method are generally stable but still exhibit slight oscillations. On the other hand, the LCR simulations exhibit more pronounced oscillations, while the LCR + DEVSS-ω and LCR + coupling simulations remain stable, displaying similar trends. All simulations achieve stability at β = 0.3. Figure 7 illustrates the simulation results at Wi = 3. At β = 0.2, oscillations occur in all algorithms, with similar stability performance observed in LCR + DEVSS-ω and LCR + coupling simulations. At β = 0.3, only the simulations utilizing LCR alone exhibit slight oscillations, while the other algorithms achieve stability. Finally, at β = 0.4, all the algorithms generally achieve stability, but the LCR cases still exhibit minimal oscillations. Regardless of the algorithm employed, an increase in the Wi number corresponds to an increase in the minimum stable viscosity ratio β. This can be attributed to the strengthening of the elastic effect of the fluid within the cavity as the Wi number increases and β decreases. Consequently, achieving stability in numerical simulations becomes more challenging.

The addition of different momentum equation stability algorithms to the LCR method has varying degrees of stabilizing effects on the simulation of low viscosity ratio driven flows in the cavity, enhancing the coupling between velocity and stress tensor. In simulations at different Wi numbers, the coupling method achieves the lowest β for stability, while the DEVSS-ω method slightly lags behind the coupling method in the simulation results under Wi = 1 conditions. However, in simulations under Wi = 2 and Wi = 3 conditions, the stability performance of the two methods is similar. The BSD method can achieve stability in simulations with medium to low viscosity ratios, but its stability is weaker than the first two methods in simulations with even lower viscosity ratios.

The incorporation of various momentum equation stability algorithms into the LCR method has different levels of stabilizing effects on the simulation of lid-driven flows in a cavity with low viscosity ratios, particularly by enhancing the coupling between velocity and stress tensor. In simulations conducted at different Wi numbers, the coupling method exhibits the lowest β value required for stability. The DEVSS-ω method closely follows the coupling method in terms of stability performance, particularly evident in simulations under Wi = 1 conditions. However, in simulations conducted under Wi = 2 and Wi = 3 conditions, the stability performance of the two methods is comparable. The BSD method demonstrates the ability to achieve stability in simulations with medium to low viscosity ratios. Nevertheless, its stability is weaker compared to the coupling and DEVSS-ω methods in simulations with even lower viscosity ratios.

In this simulation, which belongs to a transient problem, it is necessary to set an appropriate time step. When the time step in the simulation exceeds a certain value, the results will become unstable or even diverge. This critical time step is referred to as the maximum critical time step. To further compare the stability of the LCR method combined with other stabilization algorithms, this study compared the maximum critical time steps required to achieve steady-state calculations under different Wi and β conditions. The results in

Table 3. Critical time step Δt_cr/s for achieving stability by using different algorithms with different Wi and β.

	Wi = 1					Wi = 2				Wi = 3
	β = 0.1	β = 0.2	β = 0.3	β = 0.4	β = 0.5	β = 0.2	β = 0.3	β = 0.4	β = 0.5	β = 0.3	β = 0.4	β = 0.5
LCR	1 × 10−4	3 × 10−3	4.3 × 10−3	4.8 × 10−3	5 × 10−3	none	1.7 × 10−3	1.8 × 10−3	3 × 10−3	none	none	1.3 × 10−3
LCR + BSD	4 × 10−4	1 × 10−3	1.5 × 10−3	2 × 10−3	2.4 × 10−3	none	6 × 10−4	8 × 10−4	1.5 × 10−3	none	5 × 10−4	6 × 10−4
LCR + coupling	8 × 10−4	1.8 × 10−3	2.2 × 10−3	2.6 × 10−3	3 × 10−3	1.1 × 10−3	1.3 × 10−3	1.9 × 10−3	2.2 × 10−3	6 × 10−4	8 × 10−4	9 × 10−4
LCR + DEVSS-ω	1.2 × 10−3	1.9 × 10−3	2.2 × 10−3	2.6 × 10−3	3 × 10−3	9 × 10−4	1.3 × 10−3	1.8 × 10−3	2.1 × 10−3	7 × 10−4	8 × 10−4	9 × 10−4

Note: ‘none’ means that numerical calculations cannot achieve stability under the corresponding conditions.

demonstrate that as β increases, the required time step to achieve stability also increases. Similarly, under the same β, as the Wi number increases, the required time step to attain stability decreases. This indicates that smaller time steps are necessary to achieve stability in simulations conducted under high Wi number and low β conditions. Comparing the cases using the LCR method alone to the cases incorporating momentum equation stability algorithms, it is observed that the latter require smaller time steps. This difference is particularly pronounced at Wi = 1 (low Wi number conditions) but becomes less significant as the Wi number increases. The presence of minimal oscillations in the LCR and LCR + BSD cases could be attributed to the increased Wi and decreased β, which accentuate the elastic effects of the fluid. The reduced ellipticity of the momentum equation makes the coupling between velocity and stress more challenging, imposing stricter requirements on the time step.

On the other hand, the cases incorporating stability algorithms exhibit improved equation stability, especially under low viscosity ratio conditions, so the reduction in time step size is less significant at higher Wi numbers compared to the former case. When using only the LCR method to simulate conditions close to the critical viscosity ratio, the critical time step decreases sharply, making the simulation more challenging.

3.4.2. Analysis of Low Viscosity Ratio Flow Characteristics

Many studies have conducted numerical simulations of cavity-driven flows [6,20,21,22,23]. On one hand, these simulations serve as a benchmark to validate the stability and accuracy of different algorithms. On the other hand, they investigate the flow characteristics of cavity-driven flows under different conditions. However, most of these studies focus on varying the Wi number and lack simulation results for cavity-driven flows under low viscosity ratio conditions. Building upon this foundation, this section uses the LCR + coupling method to simulate cavity-driven flows and explores the influence of viscosity ratio on the flow characteristics.

To understand the flow characteristics of steady-state flow in low viscosity ratio cavity-driven flows, numerical simulations were conducted with different viscosity ratios at Wi = 1, 2, and 3 conditions. Figure 8 shows the velocity distribution along the line x/H = 0.5 and y/H = 0.75 for different β values at Wi = 1. From the figure, it can be observed that as β decreases, the minimum value of the horizontal velocity component decreases and shifts towards the top wall. Additionally, the extrema of the vertical velocity component decrease as β decreases, with the minimum and maximum values gradually approaching zero. This trend indicates that as β decreases, the contribution of elasticity increases, resulting in a decrease in the minimum value of u/U and a limitation on the circulation flow within the cavity. The variation of u/H in the central region becomes more pronounced as β increases, corresponding to the trend of the primary vortex in Figure 9. As the fluid elasticity increases with decreasing β, the asymmetry of the flow field increases, leading to significant velocity variations near the vortex, which in turn affects the velocity distribution.

Figure 8 also presents the distribution of logarithmic stress components along the lines x/H = 0.5 and y/H = 0.75. θ_xy and θ_xx reach their maximum values in the upper-right corner of the adjacent cavity. The maximum values of θ_xy and θ_xx increase with an increase in β. With an increasing in β, the variation of θ_xx becomes more pronounced in the central region, while θ_xx near the top wall increases sharply, indicating the presence of a stress boundary layer near the top wall. The distribution of θ_xx along x = 0.5 shows the same trend, and the simulation results for different viscosity ratios near the top wall are essentially the same. As β increases, the maximum value of θ_xy increases, and there is a significant variation in θ_xy near the right wall, indicating that the change in viscosity ratio has little effect on the change of elastic stress near the boundary.

Figure 10a,c illustrates the velocity distribution along the lines x/H = 0.5 and y/H = 0.75 for different β values at Wi = 2. The overall trend is similar to that observed of Wi = 1, but there are more pronounced fluctuations in the velocity distribution within the central region, which become more significant with increasing β. As β increases, the velocity extrema under Wi = 2 conditions become larger. As the Wi increases, the influence of elastic forces in the fluid becomes more significant. On the other hand, an increase in β affects the elastic contribution in the fluid, resulting in a decrease in the magnitude of the elastic forces. Figure 10b,d shows the distribution of logarithmic stress components θ_xx along the line x/H = 0.5 and θ_xy along the line y/H = 0.75, respectively. The variation trend of θ_xx along x/H = 0.5 is more pronounced. Figure 11 indicates that as β decreases, the position of the vortex core in the flow field tends to move towards the upper-left corner.

At Wi = 3, due to the occurrence of instability under low viscosity ratio conditions, Figure 12 only depicts the distribution of the velocity component u/U, θ_xx, along the line x/H = 0.5, and the distribution of the velocity component v/U, θ_xy, along the line y/H = 0.75 for β values of 0.3, 0.4, and 0.5. Under the same viscosity ratio, as the Wi number increases, the influence of fluid elasticity becomes stronger, resulting in more pronounced variations in the flow field, and the extrema of velocity and the tensor values correspondingly increase. In the case of decreasing viscosity ratio, the increase in elastic contribution leads to relatively smoother variations. This corresponds to the trend observed in the previous discussion. From Figure 13, it can be observed that as β decreases, the vortex core in the flow field slowly moves towards the upper-left corner, and the streamlines near the downstream corner become more curved.

This study investigates the influence of β on viscoelastic flow. Figure 14 displays the temporal evolution of the average kinetic energy E_k for different β values. Under the Wi = 1, β = 0 (UCM fluid) condition, E_k undergoes drastic variations between 0 s and 8 s but eventually tends towards stability. In the other conditions with different β values, the trends of E_k variations are similar. The final stable values of E_k differ for different β values. The variation of β does not significantly affect the early stage increase in kinetic energy until the peak is reached. As β decreases, the stable value of kinetic energy E_k also decreases. Furthermore, under the same β value, the stable value of average kinetic energy E_k decreases with increasing Wi number. This indicates that the contribution of viscous forces promotes an increase in the average kinetic energy of the flow field.

4. Conclusions

This study compares the stability and flow characteristics of the Log-Conformation Representation (LCR) method combined with Both-Sides-Diffusion (BSD), coupling, and DEVSS-ω stabilization algorithms for two-dimensional cavity-driven flow under low viscosity ratio conditions. The numerical solutions are compared with the standard Oldroyd-B constitutive model cavity-driven flow cases at different Wi numbers (β = 0.5) to validate the accuracy of the LCR method. The results of the cases using different momentum equation stabilization algorithms are found to be similar to those of the LCR method, demonstrating the effectiveness of these different momentum stabilization algorithms in numerical simulations under standard viscosity ratio conditions.

Using momentum equation stabilization algorithms in low viscosity ratio cavity-driven flow allows for the numerical simulation of the lowest achievable viscosity ratio. Compared to the original LCR method, the viscosity ratio is reduced by 0.03 to 0.15. The reduction in viscosity ratio increases with the increase of Wi numbers. Among the momentum equation stabilization algorithms, the coupling algorithm exhibits the best stability at low Wi numbers (Wi = 1) and can achieve stability even at the limit of zero viscosity ratio (β = 0). At Wi = 2 and 3, the lowest viscosity ratios achieved are 0.2 and 0.25, respectively. The stability of the cases using the coupling algorithm is comparable to those using the DEVSS-ω algorithm.

Under the same low viscosity ratio conditions, the critical time step of the cases using the BSD algorithm is 33% to 46% of the original LCR method. The time steps required for the coupling and DEVSS-ω algorithms are 51% to 100% of the original LCR method, with slight differences observed in some cases.

As the viscosity ratio decreases, the contribution of fluid elasticity increases, leading to enhanced asymmetry within the flow field. This results in more intricate velocity and stress variation curves in the central region of the cavity. However, the viscosity ratio has limited effect on the stress boundary layer at the top cover and corners. The average kinetic energy initially increases with the rise in velocity at the top cover and reaches a maximum after the acceleration phase. Under conditions with the same Wi number (i.e., the same fluid relaxation time), the average kinetic energy increases as the viscosity ratio decreases, corresponding to the increased contribution of fluid elasticity, until a steady state is reached.