Strong Viscous-inviscid Interaction Research Articles

AN EXPLICIT FINITE VOLUME SPATIAL MARCHING METHOD FOR REDUCED NAVIER-STOKES EQUATIONS

SUMMARY This paper develops a spatial marching method for high-speed flows based on a finite volume approach. The method employs the reduced NavierStokes equations and a pressurr splitting in the streamwise direction based on the Vigneron shategy. For marching from an upstream station to one downstream the modified five-level RungeKutta integration scheme due to Jameson and Schmidt is used. In addition, for shock handling and for good convergence properties the method employs a matrix form of the artificial dissipation terms, which has been shown to improve the accuracy of predictions. To achieve a fast rate of convergence, a local time-stepping concept is used. The method retains the time derivative in the governing equations and the solution at every spatial station is obtained in an iterative manner. The developed method is validated against two test cases: (a) supersonic flow past a flat plate; and (b) hypersonic flow past a compression comer involving a strong viscous-inviscid interaction. The computed wall pressure and wall heat transfer coefficients exhibit good general agreement with previous computations by other investigators and with experiments.

Interaction of spanwise vortices with a boundary layer

The interaction of a spanwise vortex with a boundary layer has been numerically simulated using a fractional-step method. The incompressible Navier-Stokes equations are solved to accurately predict the strong viscous–inviscid interaction between a vortex either near or embedded within a boundary layer of comparable size. A strong vortex induces an eruption and the production of a secondary vortex. The secondary vortex causes the primary vortex to rebound, a response observed in many previous experiments and numerical simulations. However, weaker vortices as well do not follow the inviscid trajectory despite the absence of a secondary vortex. Rather than creating vorticity at the wall, a weaker vortex mainly redistributes the vorticity of the boundary layer, pulling it away from the wall. The redistributed vorticity alters the path of the vortex. In the laminar cases studied the decay of the vortex is not significantly altered by the boundary layer.

Method of solution of the hypersonic boundary layer equations in the case of strong viscous-inviscid interaction

A method of solving the boundary layer equations is developed taking into account the strong interaction between the boundary layer and the outer hypersonic inviscid flow. The method is aimed at solving problems whose salient feature is the possible upstream propagation of disturbances over distances comparable with the body length. The procedure for fitting a self-consistent contour of the “effective” body using an artificially formulated boundary value problem for an ordinary second-order differential equation, which lies at the basis of the method, is considered in detail. The method is applied to the problem of flow around a flat plate with roughness in the form of an embankment or a trench; the calculated results are presented.

Close interaction between a vortex filament and a rigid sphere

The evolution of a linear vortex filament close to a rigid sphere is investigated at high Reynolds number. The limiting evolution in an ideal flow, is analysed using a cutoff method and the results are compared with those of a singular vortex approach able to account for a viscous effect on the vortex structure evolution. The computed results show the creation of a closed vortex structure in ideal flow and also, at low Reynolds number, an unrealistic reattachment of the vortex to the surface of the body. The nature of the boundary-layer development, when the no-slip condition is satisfied, is calculated near the symmetry plane. The solutions show the development of an unsteady, vortex-driven, separating boundary layer with the three-dimensional separations dependent on the initial distance of the filament from the wall. All the solutions ultimately show a rapid growth of the secondary vorticity field near the surface and suggest an ejection from the boundary layer, followed by a strong viscous-inviscid interaction.

Computation of a thermal boundary layer including strong viscous-inviscid flow interaction

Computation of a thermal boundary layer including strong viscous-inviscid flow interaction

Viscous-inviscid analysis of dual-jet ejectors

A viscous-invisc id interaction method is developed to predict the performance of two-dimension al, incompressible flow ejectors that use two primary jets. The flowfield is subdivided into a viscous region that contains the turbulent jets and an inviscid region that contains the ambient fluid drawn into the device. The inviscid flow is modeled with a higher-order panel method, and an integral method is used for the description of the viscous part. The strong viscous-inviscid interaction between the jets and the ambient fluid is simulated in an iterative process where the two regions influence each other en route to a converged solution. This formulation retains much of the essential physics of the problem but at the same time requires only a small amount of computing effort. The model is applied to a parametric study that illustrates the connection between the details of the ejector geometry and the level of thrust augmentation. The advantages of a dual-jet configuration are assessed through a comparison of the present results with similar calculations for an otherwise identical single-jet ejector.

A numerical method for treating strongly interactive three-dimensional viscous—inviscid flows

Abstract An alternative numerical method is presented for solving the three-dimensional triple-deck equations for flows with strong viscous—inviscid interaction. This method is useful in computing 3-D nonlinear flows in which surface integral conditions such as those used in previous finite difference methods or the spectral method are not feasible. The method works well when applied to a variety of obstacle shapes and it agrees favorably with solutions to interacting flows obtained by other numerical methods.

The boundary layer due to a three-dimensional vortex loop

The nature of the boundary layer induced by the motion of a three-dimensional vortex loop towards a plane wall is considered. Initially the vortex is taken to be a ring approaching a plane wall at an angle of attack in an otherwise stagnant fluid; the ring rapidly distorts into a loop shape due to the influence of the wall and the trajectory is computed from a numerical solution of the Biot-Savart integral. As the vortex loop moves, an unsteady boundary-layer flow develops on the wall. A method is described which allows the computation of the flow velocities on and near the symmetry plane of the vortex loop within the boundary layer. The computed results show the development of a variety of complex three-dimensional separation phenomena. Some of the solutions ultimately show strong localized boundary-layer growth and are suggestive that a boundary-layer eruption and a strong viscous-inviscid interaction will be induced by the moving vortex.

Viscous-inviscid analysis of transonic and low Reynolds number airfoils

A method of accurately calculating transonic and low Reynolds number airfoil flows, implemented in the viscous-inviscid design/analysis code ISES, is presented. The Euler equations are discretized on a conservative streamline grid and are strongly coupled to a two-equation integral boundary-layer formulation, using the displacement thickness concept. A transition prediction formulation of the e type is derived and incorporated into the viscous formulation. The entire discrete equation set, including the viscous and transition formulations, is solved as a fully coupled nonlinear system by a global Newton method. This is a rapid and reliable method for dealing with strong viscous-inviscid interactions, which invariably occur in transonic and low Reynolds number airfoil flows. The results presented demonstrate the ability of the ISES code to predict transitioning separation bubbles and their associated losses. The rapid airfoil performance degradation with decreasing Reynolds number is thus accurately predicted. Also presented is a transonic airfoil calculation involving shock-induced separation, showing the robustness of the global Newton solution procedure. Good agreement with experiment is obtained, further demonstrating the performance of the present integral boundary-layer formulation.

The impact of a vortex ring on a wall

The flow induced by a vortex ring approaching a plane wall on a trajectory normal to the wall is investigated for an incompressible fluid which is otherwise stagnant. The detailed characteristics of the interaction of the ring with the flow near the surface have been observed experimentally for a wide variety of laminar rings, using dye in water to visualize the flow in the ring as well as near the plane surface. Numerical solutions are obtained for the trajectory of the ring as well as for the unsteady boundary-layer flow that develops on the wall. The experimental and theoretical results show that an unsteady separation develops in the boundary-layer flow, in the form of a secondary ring attached to the wall. A period of explosive boundary-layer growth then ensues and a strong viscous-inviscid interaction occurs in the form of the ejection of the secondary vortex ring from the boundary layer. The primary ring then interacts with the secondary ring and in some cases was observed to induce the formation of a third, tertiary, ring near the wall. The details of this process are investigated over a wide Reynolds number range. The results clearly show how one vortex ring can produce another, through an unsteady interaction with a viscous flow near the wall.

Global relaxation procedure for compressible solutions of the steady-state euler equations

Transonic flow solutions of the steady-state Euler equations are obtained based on the global relaxation procedure developed by Rubin and Lin for the Reduced Navier-Stokes (RNS). The same code written for the viscous RNS equations is used to predict inviscid flows by neglecting the viscous term in these equations and deleting the no-slip boundary condition. Results for compressible flows have been computed without any stability limitation or need for explicit artificial viscosity or compressibility. A variety of solutions is presented in this paper. This method provides an alternative approach to solving the Euler equations for certain cases and for capturing of simple imbedded shock waves; although, the primary purpose of the present study is to demonstrate the applicability of this procedure for accurately representing inviscid behavior when the full RNS system is solved. The global relaxation procedure considered here has been developed primarily for solutions of the RNS equations describing flows in which strong viscous-inviscid interaction occurs. Although for the Euler equations viscous-inviscid interaction is not present, the same methodology is still applicable since the boundary value nature in these two sets of equations is predominantly associated with the unknown axial pressure gradient term (Px). Note that the RNS equations contain an exact (Euler) representation of the inviscid flowfield and the presence of the viscous term does not alter this fact. The boundary value character associated with the Px term is easily determined for the steady-state Euler equations (characteristic analysis), and is a result of wave (elliptic) propagation when this term is treated as unknown. Rubin and Lin [1, 21] presented a global relaxation technique for this elliptic problem by using a simple difference form for the numerical approximation of Px. Using boundary-layer type differencing for the convective terms, the method requires a downstream pressure boundary condition which is a characteristic of the boundary value problem. The resulting procedure consists of several forward marching sweeps for pressure relaxation. This procedure has been applied to many complex subsonic and transonic flows. With this simple treatment of the px term, the method is capable of capturing smooth imbedded shock waves, or smooth detached shock waves with imbedded subsonic regions. These results have been presented in Refs [2-5]. Application of the global relaxation procedure to inviscid flows has been demonstrated for incompressible flow over a boattail afterbody in Ref. [3]. This paper will focus on inviscid subsonic, transonic and supersonic solutions of the steady-state Euler equations, with a modified treatment of the Px term to include the compressibility effect at high Math number. The numerical method has been previously developed for viscous compressible flows. The viscous code developed in Ref. [5] is used to compute the inviscid solutions by ignoring the viscous term in the axial momentum equation and the associated no-slip boundary condition. The compressibility correction employed for the px term is based on the results obtained from the eigenvalue analysis by Vigneron et al. [6] and the stability

Calculation of supersonic flows with strong viscous-inviscid interaction

A procedure for the calculation of strong viscous-inviscid interactions in two-dimension al laminar super- sonic flows with and without separation is described. The equations solved are the so-called parabolized Navier-Stokes equations. The streamwise pressure gradient term is written as a combination of a forward and a backward difference to provide a path for upstream propagation of information. Global iteration is utilized to repeatedly update the pressure field from an initial guess until convergence is achieved. The numerical scheme employed is a new alternating direction explicit (ADE) procedure, which is used as an alternative to the more difficult to program multigrid strategy to accelerate convergence. Results are presented for flows past two flat plate related bodies. HE calculation of supersonic flows in which rapid changes in geometry or boundary condition occur often requires techniques other than classical Prandtl boundary- layer theory or other solution methods suitable for weakly flows. The reason for this is that at high Reynolds number, rapid changes in geometry or boundary condition induce large changes in the properties of the bound- ary layer over a very short length scale. When this occurs, the phenomenon of upstream influence, wherein information is propagated in the upstream direction, becomes important. This type of flow is referred to as a interacting flow. Methods that treat such flows as initial-value problems are not suitable because they are boundary value in nature, in the predominant flow direction. In this paper, a procedure for two-dimensional supersonic strongly flows is developed, which utilizes the parabolized or reduced Navier-Stokes equations. We will employ the former term here. For simplicity it will be as- sumed that x and y are nearly streamwise and stream normal directions, respectively. With special treatment of the pressure gradient term dp/dx, in the streamwise momentum equation, one can develop a procedure which is well suited for calculating weakly flows.14 In order to calculate strongly flows, a means for propagating information in the upstream direction is required. This can be satisfied by utilizing a forward difference for the stream- wise pressure gradient. In subsonic flows, where upstream influence is important even if the viscous-inviscid interaction is weak, a forward difference of px has been shown by Rubin5 to allow the calculation of such flows. Lin and Rubin6 later successfully applied forward differencing of px to a supersonic flow with weak interaction. In that study, forward differencing was utilized throughout the flowfield, including the supersonic flow. In the present technique for- ward differencing of px is applied only in the subsonic por-

Calculation of two-dimensional turbulent boundary layers using the Baldwin-Lomax model

THE algebraic turbulent eddy-viscosity model of Baldwin and Lomax is critically evaluated for a class of twodimensional, nonseparated, turbulent boundary-layer flows exhibiting adverse and favorable pressure gradients, and boundary-layer bleed. The flows are computed using the twodimensional, mass-averaged, boundary-layer equations. Overall, the results obtained using the Baldwin-Lomax model compare favorably with the experimental data. Two quantities in the outer eddy-viscosity formulation are examined by comparison with the eddy-viscosity model of Cebeci and Smith. Contents The present capabilities for numerical simulation of highspeed turbulent flows are based on a variety of turbulence models. Algebraic eddy-viscosity models are still among the most popular. Certain algebraic models require the determination of the boundary-layer edge velocity and thickness, which can result in large variations in the computed outer eddy viscosity due to the uncertainty in determining the boundarylayer edge.1 Baldwin and Lomax1 have developed a two-layer, algebraic eddy-viscosity model in an attempt to eliminate this difficulty. This model has been patterned after the model of Cebeci and Smith2; however, it does not require the determination of the boundary-layer edge. The Baldwin-Lomax turbulence model was evaluated by its originators for several cases, including a shock-wave, turbulent boundary-layer interaction on a flat plate, a supersonic turbulent compression corner, and a transonic airfoil.1 Visbal and Knight3 noted that the constants Ccp and CKleb in the Baldwin-Lomax outer formulation were dependent on flow Mach number. They also identified difficulties with the determination of the length scale in the outer eddy viscosity. Similar difficulties have been encountered by other computers.4 Despite the difficulties identified with the BaldwinLomax model, and its increasing popularity and extensive application to two- and three-dimensional flows with strong viscous-inviscid interaction, a critical evaluation of this model for nonseparated, two-dimensional, turbulent boundary-layer flows has not been performed. The present investigation seeks to contribute to the evaluation of the model by critically examining its performance for a class of turbulent boundarylayer flows. Baldwin and Lomax determined the constants Ccp and CKleb appearing in the outer formulation1 by comparison with the Cebeci-Smith model for transonic, constant-pressure,

A hybrid explicit-implicit numerical algorithm for the three-dimensional compressible Navier-Stokes equations

A hybrid explicit-implicit numerical algorithm has been developed for the three-dimensional mean compressible Navier-Stokes equations. The algorithm combines the explicit finite difference algorithm of MacCormack and an implicit algorithm for the viscous sublayer and transition wall regions of the turbulent boundary layers. The algebraic turbulent eddy-viscosity model of Baldwin and Lomax is employed. A bodyoriented coordinate transformation is utilized to facilitate treatment of arbitrary flow regions. The hybrid algorithm has been vectorized on the CDC CYBER 203 computer using the SL/1 vector programming language developed at NASA Langley. The accuracy of the numerical algorithm, established previously for twodimensional flows with strong viscous-inviscid interaction (including flow separation), is validated for threedimensional flows. The algorithm is applied to the interaction of an oblique shock wave with a turbulent boundary layer in three dimensions. The computed results generally are found to be in close agreement with the experimental data. The hybrid algorithm is shown to provide a substantial improvement in computational efficiency compared to a vectorized MacCormack explicit algorithm alone.

Global PNS solutions for subsonic strong interaction flow over a cone-cylinder-boattail configuration

The solution of the semi-elliptic or so-called parabolized Navier-Stokes equations is considered for large Reynolds numbers and subsonic flows with strong pressure interaction. Flow past a cone-cylinder-boattail configuration is investigated as a prototype of strong viscous-inviscid interaction. A global boundary-layer relaxation procedure is utilized for the formulation of the discrete boundary-value problem. The resulting marching procedure does not require a sub-layer type of approximation. Furthermore, the method does not restrict the step size in the marching direction and is free from any departure effects. Solutions with large recirculation regions are calculated.

The 1976 Freeman Scholar Lecture: Some Current Research in Unsteady Fluid Dynamics

Important unsteady fluid dynamic effects occur in a wide range of modern engineering problems. A review and critical appraisal has been made of the current research activities on topics that contain essential and unique unsteady features, especially those which cannot be approximated by quasi-steady analyses. A synopsis of the main areas covered in this paper is given below. Linear potential theory is well advanced and most of the fundamental concepts are well understood. The theory has been specially adapted for engineering purposes to many complex geometries and flow environments, but its limitations are not well established in most cases. Transonic flows have received considerable attention in recent years, and the profusion of numerical analyses of nonlinear unsteady flows has outstripped measurements. However, new experimental investigations are underway. Numerical codes are becoming much more efficient, and efforts are being made to incorporate viscous effects into them. Unsteady boundary layers have been computed with almost no complementary experimental guidance, and this deficiency is particularly acute in the turbulent case. A major conceptual difference between steady and unsteady separation has been identified and is continuing to be studied. Unsteady stall is currently under detailed examination, and recent experiments have shed considerable new insight on the fundamental mechanisms of dynamic stall on oscillating airfoils. New attempts to treat unsteady stall as a strong viscous-inviscid interaction problem are needed. Vortex shedding from bluff bodies is difficult to predict, especially in cases where body oscillations are self-induced by the fluctuating fluid dynamic forces. Nonlinear oscillator models are limited by a lack of understanding of the basic fluid dynamic phenomena. The trailing edge condition of Kutta and Joukowski for thin airfoils has been called into question recently for unsteady flows at high frequencies or with trailing-edge separation. The correct modeling of this condition is important in predicting the fluid dynamic forces on all thin lifting surfaces that fluctuate. Considerable progress has been made in each of these subjects, but none of them has been mastered. The questions that remain unanswered pose intriguing challenges to the fluid dynamics community.