Two-dimensional Incompressible Navier-Stokes Equations Research Articles

Numerical study of pressure fluctuations on a rectangular cylinder in aerodynamic oscillation

We numerically simulate unsteady flow fields around a forced-oscillating rectangular cylinder. The higher-order finite difference computation of the two-dimensional incompressible Navier-Stokes equations is applied to the dynamic interaction between the vortex motions and the torsional oscillation of a rectangular cylinder. In order to clarify the physical mechanism of the aerodynamic damping forces and the self-excited forces, we investigate the pressure fluctuations acting on the side surface. According to the computational flow visualization of the instantaneous vorticity, the vortices shed from the leading edge are controlled by the motion of a rectangular cylinder and the lock-on patterns of vortex shedding are clearly seen under the proper conditions.

A Numerical Study on Jet-Stream Propulsion of Oscillating Bodies

A Computational Fluid Dynamics (CFD) study on the jet-stream propulsion of rigidily oscillating and undulating bodies has been undertaken, by the unsteady solutions to the two-dimensional incompressible Navier-Stokes equations in conservative form, which are discretized with the finite volume method, using the pseudo-compressibility technique. Computational validation confirmed that the present method was capable to reasonably predict highly unsteady flows of biological problems. Numerical study on an oscillating hydrofoil (NACA 0012) reveals that there exists a problem of optimal propeller efficiency in generating the jet-stream in wake, but within a narrow region of the Strouhal numbers. Further analysis on a tadpole-shaped object and a fish-like body swimming in realistic kinematics, shows that the kinematics effectively produces a jet-stream propulsion with much higher propulsive efficiency than that of achieved by the oscillating hydrofoil. Investigation of Reynolds number effect for the undulatory swimming indicates that the propeller efficiency increases with increasing Reynolds number with no Re ceiling in generating the jet-stream.

Numerical solution of two-dimensional or axi-symmetric incompressible flow using the vorticity equations

A method to integrate the vorticity-velocity form of two-dimensional or axi-symmetric incompressible Navier-Stokes equations is described. The method employs equi-potential lines and streamlines of an inviscid flow as coordinate lines and the velocity field is determined from the vorticity distribution with boundary conditions on the normal velocity only, while the tangential velocities are used as boundary conditions for the vorticity. The results of numerical experiments on time-dependent flow past an impulsively started circular cylinder and sphere are, then, presented to demonstrate the performance of the scheme. Numerical results show that the present method is very stable and accurate.

Construction and validation of a discrete vortex method for the two-dimensional incompressible Navier-Stokes equations

The discrete vortex method is a Lagrangian technique for solving the two-dimensional Navier-Stokes equations for an incompressible Newtonian fluid. Construction of a robust numerical solver for the impulsively started flow past an arbitrary shaped body is described. The boundary conditions at the body are satisfied with a panel method that uses a novel curved element. The viscous effects are modelled using a combination of the random walk and diffusion velocity techniques. The computational cost is reduced by using a zonal decomposition algorithm for the velocity summation. The code has been validated by using experimental data and other numerical data to demonstrate that the solutions produced have accuracy comparable to those obtained using finite difference computations, without the effects of grid-based numerical diffusion.

An adaptive multigrid finite‐volume scheme for incompressible Navier–Stokes equations

AbstractAn algorithm for the solutions of the two‐dimensional incompressible Navier–Stokes equations is presented. The algorithm can be used to compute both steady‐state and time‐dependent flow problems. It is based on an artificial compressibility method and uses higher‐order upwind finite‐volume techniques for the convective terms and a second‐order finite‐volume technique for the viscous terms. Three upwind schemes for discretizing convective terms are proposed here. An interesting result is that the solutions computed by one of them is not sensitive to the value of the artificial compressibility parameter. A second‐order, two‐step Runge–Kutta integration coupling with an implicit residual smoothing and with a multigrid method is used for achieving fast convergence for both steady‐ and unsteady‐state problems. The numerical results agree well with experimental and other numerical data. A comparison with an analytically exact solution is performed to verify the space and time accuracy of the algorithm.

A numerical study on the flow around flat plates at low Reynolds numbers

The unsteady viscous flow around flat plates is investigated through a finite-difference computation of the incompressible two-dimensional Navier-Stokes equations. The side ratio of the plates d/h ranges from 3 to 10 with Reynolds numbers Re equal to 200 and 400, where d is the depth, h is the height, and Re is based on h. It was pointed out in our previous experiment[1] performed at Re=1000 that vortex shedding from the plates is characterized by the impinging-shear-layer instability, since the Strouhal number based on d increases stepwise with increasing d/h, as was confirmed in our numerical analysis[2]. In the present paper, the flows at two lower Reynolds numbers, Re=200 and 400, are numerically analyzed and compared with the results for Re=1000.

A numerical study on the flow around flat plates at low Reynolds numbers

The unsteady viscous flow around flat plates is investigated through a finite-difference computation of the incompressible two-dimensional Navier-Stokes equations. The side ratio of the plates d/h ranges from 3 to 10 with Reynolds numbers Re equal to 200 and 400, where d is the depth, h is the height, and Re is based on h. It was pointed out in our previous experiment[1] performed at Re=1000 that vortex shedding from the plates is characterized by the impinging-shear-layer instability, since the Strouhal number based on d increases stepwise with increasing d/h, as was confirmed in our numerical analysis[2]. In the present paper, the flows at two lower Reynolds numbers, Re=200 and 400, are numerically analyzed and compared with the results for Re=1000.

A row of counter-rotating vortices

In 1967, Stuart [J. Fluid Mech. 29, 417 (1967)] found an exact nonlinear solution of the inviscid, incompressible two-dimensional Navier–Stokes equations, representing an infinite row of identical vortices which are now known as Stuart vortices. In this Brief Communication, the corresponding result for an infinite row of counter-rotating vortices, i.e., a row of vortices of alternating sign, is presented. While for Stuart’s solution, the streamfunction satisfied Liouville’s equation, the streamfunction presented here satisfies the sinh–Gordon equation [Solitons: An Introduction (Cambridge U.P., London, 1989)]. The connection with Stuart’s solution is discussed.

Viscous flow calculations in turbomachinery channels

An implicit procedure based on the artificial compressibility formulation is presented for the numerical solution of the two-dimensional incompressible steady Navier-Stokes equations in the presence of large separated regions. Turbulence effects are accounted for by the Chien low Reynolds number form of the K - e turbulence model and the Baldwin-Lomax algebraic expression for turbulent viscosity. The governing equations are written in conservative form and irnplicitly solved in fully coupled form using the approximate factorization technique. Preliminary tests were carried out in a laminar flow regime to check the accuracy and stability of the method in two-dimensional and cylindrical axisymmetric flow configurations. After testing in laminar and turbulent flow regimes and comparing the two turbulence models, the code was successfully applied to an actual gas turbine diffuser at low Mach numbers.

Numerical Simulation of Flows Driven by a Torsionally Oscillating Lid in a Square Cavity

Numerical studies were made of the flow of a viscous fluid in a two-dimensional square container. The flows are driven by the top sliding wall, which executes sinusoidal oscillations. Numerical solutions were acquired by solving the time-dependent, two-dimensional incompressible Navier-Stokes equations. Results are presented for wide ranges of two principal physical parameters, i.e., Re, the Reynolds number and ω′, the nondimensional frequency of the lid oscillation. Comprehensive details of the flow-structure are presented. When ω′ is small, the flow bears qualitative similarity to the well-documented steady driven-cavity flow. The flow in the bulk of cavity region is affected by the motion of the sliding upper lid. On the contrary, when ω′ is large, the fluid motion tends to be confined within a thin layer near the oscillating lid. In this case, the flow displays the characteristic features of a thin-layer flow. When ω′ is intermediate, ω′ ~ O(1), the effect of the side walls is pronounced; the flow pattern reveals significant changes between the low-Re and high-Re limits. Streamline plots are constructed for different parameter spaces. Physically informative interpretations are proposed which help gain physical insight into the dynamics. The behavior of the force coefficient Cf has been examined. The magnitude and phase lag of Cf are determined by elaborate post-processings of the numerical data. By utilizing the wealth of the computational results, characterizations of Cf as functions of Re and ω′ are attempted. These are in qualitative consistency with the theoretical predictions for the limiting parameter values.

Active control of the hydraulic forces of a body by a splitter plate

The present article investigates numerically the structure of the wake behind a square-section cylinder mounted with a splitter plate. The two-dimensional incompressible Navier-Stokes equation is solved numerically. The interaction between the transverse flow (behind the body) and the splitter has considerable influence on the rearrangement of the vortex street. With the splitter plate suitably located, the peak of the lift coefficient is reduced to 40% of the isolated body case. Also, the frequency of the variation of the lift coefficient becomes smaller. It is found that there is an optimal location for the splitter plate.

Numerical Analysis of Two-Dimensional Flows around Elliptic Wings above a Flat Plate.

This paper reports on flows around two-dimensional elliptic wings which are linearly arranged above a flat plate. To solve two-dimensional time-dependent incompressible Navier-Stokes equations which use a body-fitted curvilinear coordinate system, a numerical solution method has been developed, and the method has been applied to the flows around 25% thickness elliptic wing cascades slanted at an angle of attack 15°at a Reynolds number of 1.6×103. Those wings are linearly arranged at an equal distance above a flat plate. This paper presents a number of numerical solutions for streamlines, vorticity profiles, lift, drag and pressure distributions. The lift and the drag almost regularly fluctuate with time, and the flow rates through each passage also regularly fluctuate. The present studies have been conducted as basic research into the mechanism of blowing snow off a road.

Prediction of Cascade Performance Using an Incompressible Navier–Stokes Technique

A fully elliptic, control volume solution of the two-dimensional incompressible Navier–Stokes equations for the prediction of cascade performance over a wide range is presented in this paper. The numerical technique is based on a new pressure substitution method. A Poisson equation is derived from the pressure-weighted substitution of the full momentum equations into the continuity equation. The analysis of a double circular arc compressor cascade is presented, and the results are compared with the available experimental data at various incidence angles. Good agreement is obtained for the blade pressure distribution, boundary layer and wake profiles, skin friction coefficient, losses and outlet angles. Turbulence effects are simulated by the low-Reynolds-number version of the k–ε turbulence model.

Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme, based on flux-difference splitting, is used to compute the convective terms. The upwind differencing is biased, based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. Both third-order and fifth-order differencing schemes are used on the convective fluxes throughout the grid's interior. The equations are solved using an implicit line relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudo-time, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, and good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at —90° angle of attack is computed.

Numerical simulation of natural convection in a square cavity using vorticity-velocity formulation.

Two-dimensional incompressible Navier-Stokes equations in vorticity-velocity formulation and the energy equation for temperature have been solved using a method of lines approach. The spatial derivatives are discretized by means of the central finite differential approximation, and the resulting system of the ordinary differential equations in time is integrated by a rational Runge-Kutta (RRK)scheme. Poisson equations for velocity are solved by a Group Explicit Iterative (GEI) method, using the multigrid technique. The GEI method is a fully explicit scheme so it takes advantage of the supercomputer's architecture. As the verification of the scheme, the classical problem of natural convection in a square cavity of Boussinesq fluid has chosen. Numerical calculations are obtained at a Prandtle number of 0.71 corresponding to air and Rayleigh number up to 108. The results of calculation essentially agree well with those of de Vahl Davis.

Vorticity boundary conditions and boundary vorticity generation for two-dimensional viscous incompressible flows

In this paper we present boundary conditions appropriate for the vorticity formulation of the two-dimensional incompressible viscous Navier-Stokes equations. These boundary conditions are incorporated into a finite difference scheme and the resulting method is of the “vorticity creation” type; i.e., vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied. The results of computations with this finite difference method are presented for flow past a circular cylinder. A difference scheme and computational results for a model problem, the Prandtl boundary layer equations describing flow over a semi-infinite flat plate are also presented.

Comparison of numerical methods for the calculation of two-dimensional turbulence

The estimate derived by Henshaw, Kreiss, and Reyna for the smallest scale present in solutions of the two-dimensional incompressible Navier-Stokes equations is employed to obtain convergent pseudospectral approximations. These solutions are then compared with those obtained by a number of commonly used numerical methods. If the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero, the "chopped" solution differs considerably from the convergent solution, even at early times. In the case that the regular viscosity is replaced by a hyperviscosity term, i.e., the square of the Laplacian, we also derive an estimate for the smallest scale present. If the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution, the hyperviscosity solution is also completely different from the convergent solution. If we "tune" the hyperviscosity coefficient, then the solutions are similar in amplitude or phase, but not both. The solution obtained by a second-order difference method with twice the number of points as the pseudospectral model, or a fourth-order difference method with the same number of points as the pseudospectral model, is essentially identical to the convergent solution. This is reasonable since most of the energy of the solution is contained in the lower part of the spectrum.

ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS USING NONSTAGGERED GRIDS

Pressure boundary conditions satisfying the normal momentum equation at solid boundaries with second-order accuracy are developed. Implementation of these conditions in an explicit numerical procedure for the two-dimensional incompressible Navier-Stokes equations enables convergent and accurate solutions for the driven cavity problem provided that the integral constraint of the Neumann boundary conditions is satisfied.

A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations

A new Chebyshev pseudospectral technique (based on the projection method that was previously applied by the authors to the solution of two-dimensional incompressible Navier-Stokes equations in primitive variables for nonperiodic boundary conditions) is extended to solve the three-dimensional Navier-Stokes equations. The crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain, on the boundaries as well as in the interior. The key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure Poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition. The method was tested on a two-dimensional driven cavity flow and the results were compared with those of the most accurate finite-difference calculation. The three-dimensional driven cavity flow was then calculated at the same Reynolds numbers as the two-dimensional cases, i.e., Re = 100, 400, and 1000. In the calculated results, three-dimensional boundary effects were observed in all cases and became more apparent with increasing Reynolds number.

A method for computing flow fields around moving bodies

A new method for computing flow fields with arbitrarily moving boundaries is proposed. Under the concept of Lie derivatives the field equations in general moving coordinates are derived, which consist of several kinds of equations, for example, one written in Viviand's conservative form. According to our formulation, it is natural and reasonable to consider that the computational coordinates fitted to the body move in space, contrary to the usual computational procedures. The two-dimensional incompressible Navier-Stokes equations in gemeral moving coordinates are solved by a finite difference method. The present calculations are made for (a) the blood flow in human ventricle, and (b) the dynamic stall process on oscillating airfoil. Consequently it is shown that the flows generated by moving bodies can easily be analyzed by the present method.