Momentum Interpolation Research Articles

Finite volume simulation of viscoelastic laminar flow in a lid-driven cavity

A finite volume technique is presented for the numerical solution of steady laminar flow of Oldroyd-B fluid in a lid-driven square cavity over a wide range of Reynolds and Weissenberg numbers. Second order central difference (CD) scheme is used for the convection part of the momentum equation while first order upwind approximation is employed to handle viscoelastic stresses. A non-uniform collocated grid system is used. Momentum interpolation method (MIM) is used to evaluate face velocity. Coupled mass and momentum conservation equations are solved through iterative SIMPLE (semi-implicit method for pressure-linked equation) algorithm. Detailed investigation of the flow field is carried out in terms of velocity and stress fields. Differences between the behavior of Newtonian and viscoelastic fluids, such as the normal stress effects and secondary eddy formations, are highlighted.

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow (Power-Law model). The collocated (i.e. non-staggered) arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

Implementation of an efficient segregated algorithm-IDEAL on 3D collocated grid system

The segregated algorithm-IDEAL (inner doubly-iterative efficient algorithm for linked-equations) is an efficient and stable algorithm. In this algorithm, there exist inner doubly-iterative processes for pressure equation, which almost completely overcome two approximations in SIMPLE algorithm. Thus the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of iteration process. In this paper, implementation of the IDEAL algorithm on a 3D collocated grid system is conducted. The interface velocity is calculated by the modified momentum interpolation method (MMIM), by which the converged result is independent of the under-relaxation factor. Finally, five three-dimensional incompressible fluid flow and heat transfer problems are provided to compare the convergence rate and robustness between the IDEAL and three other most widely-used algorithms (SIMPLER, SIMPLEC and PISO). By the comparison it can be concluded that the IDEAL algorithm is more robust and efficient than the three other algorithms.

A New Method for Constructing the Coefficients of Pressure Correction Equation for Colocated Unstructured Grids

One of the important equations in numerical solution of Navier Stokes equations is the pressure correction equation. In this article, a new method for constructing the coefficients of this equation for colocated unstructured meshes is proposed. This method is based on momentum interpolation. The method is compared with the approach adopted in literature. The complete discretization of the Navier Stokes equations using finite volume method is presented. An algorithm similar to SIMPLE is used to evaluate the rate of convergence on two sample problem. The results show that by using rectangular grids, two methods have the same performance, but when triangular meshes are applied, the new method increases the rate of convergence.

Numerical simulations of ice accumulation under ice cover along a river bend

In this paper, the k-έ two-equation turbulence model has been used to simulate ice accumulation under ice cover along a river bend. A 2D depth-averaged numerical model has been developed in a nonorthogonal coordinate system with nonstaggered curvilinear grids. In this model, the contravariant velocity has been treated as an independent variable. To avoid the pressure oscillation in the nonstaggered grids, the momentum interpolation has been introduced to interpolate variables at the interface. The discretization equations have been solved by using pressure correction algorithms. An equation has been developed for describing the deformation of ice jam bottom. The thickness distribution of ice accumulation (ice jam) along the bend has been simulated. The developed model has been applied to the experimental studies under different conditions carried out at the Hefei University of Technology. Results indicate that all simulated thickness of ice accumulation agrees reasonably well with the measured thickness of ice accumulation in laboratory.

A Finite-Volume Method for Navier-Stokes Equations on Unstructured Meshes

A novel finite-volume formulation is proposed for unsteady solutions on complex geometries. A computer code based on a cell-centered finite-volume method is developed to solve both two-dimensional (2-D) and three-dimensional (3-D) Navier-Stokes equations for incompressible laminar flow on unstructured grids. A collocated (i.e., nonstaggered) arrangement of variables is used. The convective terms have provision for a variable upwinding factor, and the diffusion fluxes are computed in a novel and natural way. The pressure–velocity decoupling is avoided by momentum interpolation. The method is shown to have nearly second-order accuracy even on nonorthogonal grids. Some Navier-Stokes solutions, both 2-D and 3-D, are presented to verify the method with standard benchmark solutions. The comparison of present results with those in the literature is good. A computational study of 2-D laminar flow and heat transfer past a triangular cylinder in free stream is presented for the range 10 ≤ Re ≤ 200.

Preconditioning Based on a Partially Implicit Implementation of Momentum Interpolation for Coupled Solvers

A new implicit and coupled algorithm for solving the Navier-Stokes equations is presented. A discretized Poisson-like equation for pressure is derived from the continuity equation by using expressions for the face velocities corrected with the momentum interpolation approach. A partially implicit implementation of this equation is proposed in order to achieve the coupling of velocity and pressure without increasing the size of the computational molecule. The ability of the algorithm to solve both constant- and variable-density flows is shown by presenting numerical results for the lid-driven cavity flow at moderate and high Reynolds numbers, and for the buoyancy-driven cavity flow with large temperature differences. A fully implicit scheme is used for discretizing the transient terms, and the efficiency and accuracy of the algorithm for unsteady flows is shown by solving the laminar vortex shedding over a square cylinder.

An All-Speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour

A new scheme, All-Speed-Roe scheme, was proposed for all speed flows. Compared with traditional preconditioned Roe scheme, All-Speed-Roe scheme changes non-linear eigenvalues in the numerical dissipation terms of Roe-type schemes. With an asymptotic analysis, the low Mach number behaviour of the scheme is studied theoretically in two ways. In one way, All-Speed-Roe scheme is regarded as finite magnification of Low-Speed-Roe scheme in the low Mach number limit. In the other way, a general form of All-Speed-Roe scheme is analyzed. Both ways demonstrate that All-Speed-Roe scheme has the same low Mach number behaviour as the original governing equation in the continuous case, which includes three important features: pressure variation scales with the square of the Mach number, the zero order velocity field is subject to a divergence constraint, and the second order pressure satisfies a Poisson-type equation in the case of constant-entropy. The analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned Roe scheme does not satisfy the divergence constraint as the Mach number vanishes. Moreover, the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned Roe scheme inherently possesses to suppress checkerboard decoupling. In the end, general rulers for modifying non-linear eigenvalues are obtained. Finally, several numerical experiments are provided to support the theoretical analysis. All-Speed-Roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations.

A Compact Momentum Interpolation Procedure for Unsteady Flows and Relaxation

An improved version of the momentum interpolation approach for calculating velocities at the cell faces in nonstaggered grids is proposed. The procedure is developed for unsteady flows and takes into account the inclusion of relaxation. Regarding the integration in time, the first-order Euler and the second-order Euler and Adams-Moulton schemes are analyzed. The proposed procedure results in a compact, easy-to-implement expression which is shown to provide the desirable performance: Spurious oscillations of pressure are avoided; converged, steady solutions are independent from relaxation coefficient and time-step size; and the accuracy of the discretization is not negatively affected.

A quasi-implicit time advancing scheme for unsteady incompressible flow. Part I: Validation

A quasi-implicit fractional-step method is presented for computing unsteady incompressible flow on unstructured grids. A non-staggered grid system is employed rather than a staggered grid system because of the simplicity and ease of extension to three-dimensional analysis. In this study, the momentum interpolation method, developed by Rhie and Chow [C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532] and further extended by Zang et al. [Y. Zang, R.L. Street, J.R. Koseff, A non-staggered grid, fractional-step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates, J. Comput. Phys. 114 (1994) 18–33], is applied to problems on unstructured grids to resolve the pressure oscillation problem occurring in a non-staggered grid system. An implicit time advancing scheme is used in order to remove the time step restriction and to reduce the required CPU time for problems with complex geometries. The nonlinear equations resulting from this fully implicit scheme are linearized without deteriorating the overall time accuracy. The system matrices are solved using the CG family method, known with P-BiCGSTAB, for momentum equation and P-CG for pressure Poisson equation. The present numerical method is applied to solve four benchmark problems and the results show good agreement with previous experimental and numerical results.

An Efficient and Robust Numerical Scheme for the SIMPLER Algorithm on Non-Orthogonal Curvilinear Coordinates: CLEARER

In this article, an Improved SIMPLER (CLEARER) algorithm is formulated to solve the incompressible fluid flow and heat transfer on the nonstaggered, nonorthogonal curvilinear grid system. By virtue of a modified momentum interpolation method in calculating the interface contravariant velocity in both the predictor step and the corrector step, the coupling between pressure and velocity is fully guaranteed, and the conservation law is also satisfied. A second relaxation factor is introduced in the corrector step, of which the convergent solution is independent. By setting the second relaxation factor less than the underrelaxation factor for the velocity to some extent, both the convergence rate and robustness can be greatly enhanced. Meanwhile, the CLEARER algorithm can also overcome the severe grid nonorthogonality. With the simplified pressure-correction equation, the convergent solution can still be obtained even when the intersection angle among grid lines is as low as 1°, which may provide valuable guidance in studying the fluid flow in complex geometries.

An Improved Numerical Scheme for the SIMPLER Method on NonOrthogonal Curvilinear Coordinates: SIMPLERM

In this article, an improved numerical algorithm named SIMPLERM is proposed for incompressible fluid flow computations on the nonstaggered and nonorthogonal curvilinear grid system. In the proposed algorithm, the contravariant velocities are chosen as the cell face velocities and the Cartesian components as the primary variables. The velocity under-relaxation factor is incorporated into the momentum interpolation, and special treatment is adopted to avoid the underrelaxation factor dependence of the velocity solution. In addition, a 1 − δ pressure difference is introduced into the interfacial contravariant velocity determination. Compared with the existing implementation methods of the SIMPLE family on non-staggered and nonorthogonal grids, the SIMPLERM algorithm can guarantee the coupling between velocity and pressure, underrelaxation independence of the solution, and satisfaction of the conservation law, while still possessing sufficient robustness.

Development of a flow analysis code using an unstructured grid with the cell-centered method

A conservative finite-volume numerical method for unstructured grids with the cell-centered method has been developed for computing flow and heat transfer by combining the attractive features of the existing pressure-based procedures with the advances made in unstructured grid techniques. This method uses an integral form of governing equations for arbitrary convex polyhedra. Care is taken in the discretization and solution procedure to avoid formulations that are cell-shape-specific. A collocated variable arrangement formulation is developed, i.e. all dependent variables such as pressure and velocity are stored at cell centers. For both convective and diffusive fluxes the forms superior to both accuracy and stability are particularly adopted and formulated through a systematic study on the existing approximation ones. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are computed by using a linear reconstruction based on the divergence theorem. Momentum interpolation is used to prevent the pressure checkerboarding and a segregated solution strategy is adopted to minimize the storage requirements with the pressure-velocity coupling by the SIMPLE algorithm. An algebraic solver using iterative preconditioned conjugate gradient method is used for the solution of linearized equations. The flow analysis code (PowerCFD) developed by the present method is evaluated for its application to several 2-D structured-mesh benchmark problems using a variety of unstructured quadrilateral and triangular meshes. The present flow analysis code by using unstructured grids with the cell-centered method clearly demonstrate the same accuracy and robustness as that for a typical structured mesh.

Numerical Study of Laminar Forced Convection Fluid Flow and Heat Transfer From a Triangular Cylinder Placed in a Channel

Computational study of two-dimensional laminar flow and heat transfer past a triangular cylinder placed in a horizontal channel is presented for the range 80≤Re≤200 and blockage ratio 1/12≤β≤1/3. A second-order accurate finite volume code with nonstaggered arrangement of variables is developed employing momentum interpolation for the pressure-velocity coupling. Global mode of cross-stream velocity oscillations predict the first bifurcation point increases linearly with blockage ratio with no second bifurcation found in the range of Re studied. The Strouhal number and rms of lift coefficient increase significantly with blockage ratio and Reynolds number while overall Nusselt number remains almost unchanged for different blockage ratios. At lower blockage ratios, flow is found to be similar to the unconfined flow and is more prone to wake instability. Instantaneous streak lines provide an excellent means of visualizing the von Kármán vortex street.

Analysis of heat transfer in simultaneously developing pulsating laminar flow in a pipe with constant wall temperature

Numerical studies of flow and heat transfer in a circular tube under pulsating flow condition were carried out in the laminar regime. The flow at the inlet consists of a fixed part and a pulsating component, which varies sinusoidally in time. The flow was both thermally and hydrodynamically developing while the tube wall was kept at a uniform temperature. The solution of two-dimensional Navier–Stokes equation was performed using the SIMPLE algorithm with the momentum interpolation technique of Rhie and Chow. By analysing the data generated from the simulation, it is observed that in the range of present study (frequency: 0–20 Hz; amplitude: < 1.0), pulsation has no effect on time-averaged heat transfer, although the Nu distribution varies in time in the near-entry region of the pipe.

Stability of Navier–Stokes discretizations on collocated meshes of high anisotropy and the performance of algebraic multigrid solvers

A numerical instability is identified in Navier–Stokes discretizations on meshes of high anisotropy. The instability occurs under conditions of low Reynolds number (and in the Stokes limit) for collocated-mesh discretizations based on physical (momentum) interpolation schemes. It is responsible for the poor performance reported for some algebraic multigrid solvers (previously attributed to possible deficiencies in the solvers). The problem may be alleviated by not employing uniformly anisotropic meshes. A graded stretching/compaction that leaves part of the domain spanned by elements of moderate aspect ratio can provide sufficient velocity–pressure coupling to stabilize the system. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical simulation of unconfined flow past a triangular cylinder

Numerical simulations of two-dimensional laminar flow past a triangular cylinder placed in free-stream at low Reynolds number (10⩽Re⩽250) are performed. A finite volume method, second-order accurate in space and time, employing non-staggered arrangement of the variables with momentum interpolation for the pressure–velocity coupling is developed. Global mode analysis predicts Recr=39.9 which confirms the results of earlier studies. Vortex shedding phenomena is found to be similar to the square cylinder with no second bifurcation in the range of Re studied. A discussion on the time-averaged drag coefficient, rms of lift coefficient and Strouhal number is presented. Particle tracking and the instantaneous streaklines provide an excellent means of visualizing the von Karman vortex street. Copyright © 2006 John Wiley & Sons, Ltd.

Three-dimensional numerical simulation of viscoelastic flow through an abrupt contraction geometry

AbstractA numerical scheme for simulating three-dimensional viscoelastic flow was developed. The three-dimensional finite volume method (FVM) based on a non-staggered grid and the semi-implicit method for pressure-linked equations (SIMPLE) were adopted to solve the continuity and momentum equations. As we used a non-staggered grid, the momentum interpolation method (MIM) was employed to avoid checkerboard type pressure fields. Viscoelastic properties of the fluid were described by the Phan-Thien and Tanner (PTT) model, which was treated by the compressive interface capturing scheme for arbitrary meshes (CICSAM). The elastic viscous split stress scheme (EVSS) was used to decouple the velocity and the stress fields. Algebraic equations obtained by the above schemes were handled by an iterative solver and a multi-grid method was applied to accelerate convergence. In order to verify the scheme developed in this study, Newtonian flow in a rectangular duct was studied and the resulting velocity fields were compared with the analytical flow fields. Then viscoelastic flow in 4:1 contraction geometry, one of the most frequently used benchmarking problems, was predicted by applying the fully three-dimensional finite volume method.

비정렬격자계를 사용하는 3차원 유동해석코드 개발 (I) - 수치해석방법 -

A conservative pressure-based finite-volume numerical method has been developed for computing flow and heat transfer by using an unstructured grid system. The method admits arbitrary convex polyhedra. Care is taken in the discretization and solution procedures to avoid formulations that are cell-shape-specific. A collocated variable arrangement formulation is developed, i.e. all dependent variables such as pressure and velocity are stored at cell centers. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are found by a novel second-order accurate spatial discretization. Momentum interpolation is used to prevent pressure checkerboarding and the SIMPLE algorithm is used for pressure-velocity coupling. The resulting set of coupled nonlinear algebraic equations is solved by employing a segregated approach, leading to a decoupled set of linear algebraic equations for each dependent variable, with a sparse diagonally dominant coefficient matrix. These equations are solved by an iterative preconditioned conjugate gradient solver which retains the sparsity of the coefficient matrix, thus achieving a very efficient use of computer resources.

A collocated finite volume method for incompressible flow on unstructured meshes

A finite volume method on unstructured meshes for the solution of incompressible Navier-Stokes equations is presented in this paper. New second-order spatial discretisations for convection and diffusion terms are developed. A non-staggered variable arrangement was utilised for all variables. Momentum interpolation technique was employed to avoid pressure checker boarding. To minimise storage requirement, a segregated solution strategy was adopted with pressure and velocity coupled by the SIMPLE method. The proposed finite volume scheme was applied to several standard problems. Computational results show that the performance of the scheme is satisfactory.