Numerical solution of the steady incompressible Navier—Stokes equations for the flow past a sphere by a multigrid defect correction technique

A nested non-linear multigrid algorithm is developed to solve the Navier–Stokes equations which describe the steady incompressible flow past a sphere. The vorticity–streamfunction formulation of the Navier–Stokes equations is chosen. The continuous operators are discretized by an upwind finite difference scheme. Several algorithms are tested as smoothing steps. The multigrid method itself provides only a first-order-accurate solution. To obtain at least second-order accuracy, a defect correction iteration is used as outer iteration. Results are reported for Re = 50, 100, 400 and 1000.     

ONSET OF ASYMMETRY: FLOW PAST CIRCULAR AND ELLIPTIC CYLINDERS

A computational study of the development of two- dimensional unsteady viscous incompressible flow around a circular cylinder and elliptic cylinders is undertaken at a Reynolds number of 10,000. A higher- order upwind scheme is used to solve the Navier–Stokes equations by the finite difference method in order to study the onset of computed asymmetry around bluff bodies. For the computed cases the ellipses develop asymmetry much earlier than the circular cylinder. The receptivity of the computed flows in the presence of discrete roughness and surface vibration is studied. Finally, the role of discrete roughness in triggering asymmetry for flow past a circular cylinder is studied and compared with flow visualization experiments at Re=10,000     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

Implicit unfactored implementation of two-equation turbulence models in compressible Navier–Stokes methods

An implicit unfactored method for the coupled solution of the compressible Navier–Stokes equations with two-equation turbulence models is presented. Both fluid-flow and turbulence transport equations are discretized by a characteristics-based scheme. The implicit unfactored method combines Newton subiterations and point-by-point Gauss–Seidel subrelaxation. Implicit-coupled and -decoupled strategies are compared for their efficiency in the solution of the Navier–Stokes equations in conjunction with low-Re two-equation turbulence models. Computations have been carried out for the flow over an axisymmetric bump using the k–ϵ and k–ω models. Comparisons have been obtained with experimental data and other numerical solutions. The present study reveals that the implicit unfactored implementation of the two-equation turbulence models reduces the computing time and improves the robustness of the CFD code in turbulent compressible flows. © 1998 John Wiley & Sons, Ltd.     

Finite element modified method of characteristics for the Navier–Stokes equations

An algorithm based on the finite element modified method of characteristics (FEMMC) is presented to solve convection–diffusion, Burgers and unsteady incompressible Navier–Stokes equations for laminar flow. Solutions for these progressively more involved problems are presented so as to give numerical evidence for the robustness, good error characteristics and accuracy of our method. To solve the Navier–Stokes equations, an approach that can be conceived as a fractional step method is used. The innovative first stage of our method is a backward search and interpolation at the foot of the characteristics, which we identify as the convective step. In this particular work, this step is followed by a conjugate gradient solution of the remaining Stokes problem. Numerical results are presented for:

• a Convection–diffusion equation. Gaussian hill in a uniform rotating field.
• b Burgers equations with viscosity.
• c Navier–Stokes solution of lid‐driven cavity flow at relatively high Reynolds numbers.
• d Navier–Stokes solution of flow around a circular cylinder at Re=100.

Copyright © 2000 John Wiley & Sons, Ltd.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of viscous water column collapse using adapting hierarchical grids

An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at Re=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd.     

High velocity flow through fractured and porous media: the role of flow non-periodicity

In the present paper we examine the evolution of the macroscopic flow law in a crenellated channel, representing an element of fractured or porous medium and in function of the Reynolds number Re. A numerical analysis based on the Navier–Stokes equations is applied. We focus on the influence of the flow periodicity or non-periodicity upon the macroscopic law. The physical explanation of the non-linear deviation from Darcy's law is still an issue, as the Ergun–Forchheimer law admitted for high Reynolds numbers comes up against some theoretical problems. In the periodic case, three non-linear flow regimes were revealed: a cubic flow with respect to velocity at low Re, an intermediate non-quadratic law, and a self-similar mode independent of Re at very high Re. The Forchheimer law is not confirmed. The case of a non-periodic flow clearly highlights the link between the flow non-periodicity and the quadratic law. The quadratic deviation becomes all the more important as the non-periodicity degree is high.     

Stokes eigenfunctions and Galerkin projection of the disturbance equations in plane Poiseuille flow: a systematic analytical approach

In the present paper, the L ²-normalized Stokes eigenfunctions for plane Poiseuille flow, which form an orthonormal functional basis for the space of disturbances, are written in a general exponential form. Then, the evolution equations for the disturbances are Galerkin-projected on the considered basis functions, and all the terms of the resulting dynamical system are expressed systematically in analytical form. Finally, a numerical example is given in which the proposed basis functions are used for the simulation of the time evolution of the critical disturbance predicted by the energetic stability theory.     

A NUMERICAL STUDY OF THE FLOW OVER ELLIPSOIDAL OBJECTS INSIDE A CYLINDRICAL TUBE

A numerical scheme is developed to obtain the flow field around one, two and five ellipsoidal objects inside a cylindrical tube. The scheme uses the Galerkin finite element technique and the primitive variable(u–v–p) formulation. The two-dimensional incompressible Navier–Stokes equations are solved numerically by using the direct mixed interpolation method. A Picard iteration scheme is used for the solution of the resulting system of non-linear algebraic equations. The computer code is verified by checking with known analytical solutions for the flow past a sphere. Results for the shear stress distributions along the ellipsoids, forces and drag coefficients are obtained for different geometric ratios and Reynolds numbers. Some of the intermediate computational results on the velocity fields developed are also reported.     

Axisymmetric stagnation flow obliquely impinging on a moving circular cylinder with uniform transpiration

Laminar stagnation flow, axisymmetrically yet obliquely impinging on a moving circular cylinder, is formulated as an exact solution of the Navier–Stokes equations. Axial velocity is time‐dependent, whereas the surface transpiration is uniform and steady. The impinging free stream is steady with a strain rate k?. The governing parameters are the stagnation‐flow Reynolds number Re=k?a²/2ν, and the dimensionless transpiration S=U₀/k?a. An exact solution is obtained by reducing the Navier–Stokes equations to a system of differential equations governed by Reynolds number and the dimensionless wall transpiration rate, S. The system of Boundary Value Problems is then solved by the shooting method and by deploying a finite difference scheme as a semi‐similar solution. The results are presented for velocity similarity functions, axial shear stress and stream functions for a variety of cases. Shear stresses in all cases increase with the increase in Reynolds number and suction rate. The effect of different parameters on the deflection of viscous stagnation circle is also determined. Copyright © 2010 John Wiley & Sons, Ltd.     

Suppression of vortex shedding for flow around a circular cylinder using optimal control

Adjoint formulation is employed for the optimal control of flow around a rotating cylinder, governed by the unsteady Navier–Stokes equations. The main objective consists of suppressing Karman vortex shedding in the wake of the cylinder by controlling the angular velocity of the rotating body, which can be constant in time or time‐dependent. Since the numerical control problem is ill‐posed, regularization is employed. An empirical logarithmic law relating the regularization coefficient to the Reynolds number was derived for 60?Re?140. Optimal values of the angular velocity of the cylinder are obtained for Reynolds numbers ranging from Re=60 to Re=1000. The results obtained by the computational optimal control method agree with previously obtained experimental and numerical observations. A significant reduction of the amplitude of the variation of the drag coefficient is obtained for the optimized values of the rotation rate. Copyright © 2002 John Wiley & Sons, Ltd.     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

A short note on the general boundary element method for viscous flows with high Reynolds number

In this paper, the general boundary element method and the parallel computation are employed to solve laminar viscous flows in a driven square cavity, governed by the exact Navier–Stokes equations. Using the solution at Re=0 as the initial approximation, the convergent numerical results for high Reynolds number at Re=7500 are obtained, for the first time, by the boundary element method. This verifies the validity and great potential of the general boundary element method for highly non‐linear problems, which may greatly enlarge application regions of the boundary element method in science and engineering. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of incompressible two‐phase flows with a Boussinesq–Scriven interface stress tensor

We consider the numerical simulation of a three‐dimensional two‐phase incompressible flow with a viscous interface. The simulation is based on a sharp interface Navier–Stokes model and the Boussinesq–Scriven constitutive law for the interface viscous stress tensor. In the recent paper [Soft Matter 7, 7797–7804, 2011], a model problem with a spherical droplet in a Stokes Poiseuille flow with a Boussinesq–Scriven law for the surface viscosity has been analyzed. In that paper, relations for the droplet migration velocity are derived. We relate the results obtained with our numerical solver for the two‐phase Navier–Stokes model to these theoretical relations. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of time preconditioning and high‐order compact discretization method for low Mach number flows

The paper describes a combination of a preconditioning method with a high‐order compact discretization scheme for the purpose of solving the compressible Navier–Stokes equations in moderate and low Mach number regimes. When combined with properly modified characteristic boundary conditions, the proposed approach is very efficient from the point of view of convergence acceleration and accuracy of the results. The computations were performed in typical benchmark cases including the Burggraf flow for which an analytical solution exists, the flow over a backward facing step, and also the flow in 2D and 3D shear‐driven cavities. Depending on the test case, the results were obtained for the Mach number in the range M = 0.001 ? 0.5 and the Reynolds number Re = 1 ? 1000. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulations of laminar flow over a 3D backward-facing step

A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.     

The solution of viscous incompressible jet flows using non‐staggered boundary fitted co‐ordinate methods

A new approach for the solution of the steady incompressible Navier–Stokes equations in a domain bounded in part by a free surface is presented. The procedure is based on the finite difference technique, with the non‐staggered grid fractional step method used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. In order to design an effective free solution scheme, we distinguish between flows dominated by surface tension and those dominated by inertia and viscosity. When the surface tension effect is insignificant we used the kinematic condition to update the surface; whereas, in the opposite case, we used the normal stress condition to obtain the free surface boundary. Results obtained with the improved boundary conditions for a plane Newtonian jet are found to compare well with the available two‐dimensional numerical solutions for Reynolds numbers, up to Re=100, and Capillary numbers in the range of 0≤Ca<1000. Copyright © 2001 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A postprocessing mixed finite element method for the Navier–Stokes equations

A fully discrete postprocessing mixed finite element scheme is considered for solving the time-dependent Navier–Stokes equations. In the PP method, we only consider a non-linear equation in the coarse-level subspace and a linear problem in the fine-level subspace. The analysis shows that the PP scheme can reach the same accuracy as the standard Galerkin method with a very fine mesh size h by an appropriate choice of H. Numerical examples are provided that confirm both the theoretical analysis and the corresponding improvement in computational efficiency.