Thin-layer Navier-Stokes Research Articles

Numerical investigation of separated flow around a bent-nose biconic

The purpose of this article is to investigate the separated flow around the bent-nose biconic, in which the second-order TVD (total variation diminishing) method is applied and the numerical flux is a Harten-Yee type. The governing equation is a thin layer Navier-Stokes equation. The boundary condition on the wall is the adiabatic wall. The whole flow field is laminar. The numerical conditions are as follows: Mach number 7.0, Reynolds number 8.38·10 5. The surface pressure distribution at the leeward side is a little higher in the vicinity of the symmetrical plane. This seems to be caused by the development of the vortex at the leeward side due to the separated flow. The location of a separation line and the size are in very good agreement with the experiment. Also the secondary separation is captured very clearly. The characteristics of the separated flow around the bent-nose biconic are clarified.

Vortical flow solutions using a time-lagged, thin-layer, Navier-Stokes algorithm

A proven inviscid algorithm (Program EAGLE - Flow Solver) has been modified to use the flux-difference split scheme of Roe to solve the time-lagged, thin-layer approximation to the Navier-Stokes equations. Calculations are presented evaluating the ability of the time-lagged, thinlayer Navier-Stokes algorithm to obtain transonic solutions for lifting bodies at incidence angles sufficient for the flow to be dominated by large-scale free vortices in the leeside flowfield.

Numerical investigation of airfoil/jet/fuselage-undersurface flowfields in ground effect

THE Euler/Reynolds-averaged compressible NavierStokes, continuity, and energy equations are solved in conjunction with a two-equation (K-e) turbulence model for two-dimensional flows related to vertical/shor t takeoff and landing (V/STOL) aircraft. Beam and Warming's implicit factored central-difference scheme with/without diagonalization is introduced. For the airfoil/jet/ground-interaction flowfields, the in viscid and laminar flows are considered. Different values of parameters (such as width, velocity, and temperature of jet; height of jet above the ground plane; and freestream Reynolds number) are employed to study the physical phenomena and aerodynamic coefficients. Also presented is information about the turbulent impinging jet with curved fuselageunder surf ace, pressure distributions on the ground plane and fuselage-under surf ace, and distributions of turbulent kinetic energy and velocity component along the center line. Contents The aerodynamic interaction caused by the flow around the airfoil, fuselage-under surf ace, and ground in the presence of lift jets is important in V/STOL aircraft design. Based on the solution of three-dimensi onal thin-layer Navier-Stokes equations, the velocity vectors and suck-down effect of laminar flow around a delta planform wing, equipped with two thrust reverser jets, in ground effect have been studied by Chawla et al.1 To describe the vortex formulation behind the jet and to include the effect of jet/ground interaction on the flowfield around the airfoil, Agarwal and Deese2 modified Jameson's Euler code (FLO-53) for computing almost incompressible flow. Agarwal and Bower3 studied the flow behaviors of the turbulent jet impingement flowfields with curved fuselage-undersurface. The governing equations, including a two-equation model, were expressed in stream function/vorticity forms and formulated by the finite-difference method. In this paper, Beam and Warming's scheme with/without diagonal form4'5 is introduced to solve the Euler/Reynolds-averaged compressible full Navier-Stokes, continuity, and energy equations in conjunction with a two-equation (K-e) turbulence model.6 The air foil/jet/fuselage-under surf ace flowfields in ground effect are investigated. For the air foil/jet/ground-inter action flowfields, the methods of characteristi c, average process, slip/no-slip, and adiabatic-wall conditions are respectively applied to the far-field boundary, trailing edge, wake cut, surface of airfoil, and ground plane. Velocity and temperature about the entering jet

Modeling of subsonic flow through a compact offset inlet diffuser

This paper presents a comparison of computational and experimental results that were obtained during an investigation of compressible flow through a compact, highly offset diffuser. Entrance values for the Mach number and the Reynolds number were 0.341 and 5.75(10)5, respectively. Strong curvature of the diffuser centerline produced extensive separation along one wall, which resulted in an exhaust flow with highly complex crossflow patterns and spatial nonuniformities. Our objective was to evaluate the use of an existing thin-layer Navier-Stokes code (ARC3D) to predict effects of diffuser shape and inlet flow properties on pressure recovery and exit flow quality. Comparisons are shown between computed and measured flow velocity components and wall pressure distributions. Agreement was satisfactory except in regions of separated flow. The Baldwin-Lomax algebraic turbulence model used in the code does not appear to represent adequately this complex separated region. We present preliminary results of computations made with a one-half-equation turbulence model, which accounts for some of the history effects in computing the turbulence length scale. The computed results show that the details of the axial velocity distribution in the separated flow region were quite different for the two turbulence models, but other features of the flow, such as the static pressure distributions, were only slightly affected.

Gasdynamic behaviors of reflected-shock waves in ionizing argon in a shock tube.

Phenomena by which reflected-shock waves in ionizing argon develop to bifurcated shock wave with complicated structures in a shock tube were observed. In the present investigation, finite-difference simulations are performed for such phenomena. The thin-layer Navier-Stokes equations taking account of ionization effects are numerically solved by applying a procedure consisting of the FCT scheme and the Crank-Nicolson scheme. The computational results are compared with the experimental results. The present simulations are shown to reproduce a series of visualized reflected-shock flow fields qualitatively. This fact leads us to conclude that complicated flow fields emerging after the reflection of ionizing shock waves in a shock tube are due to interactions of reflected-shock waves with ionization relaxation processes as well as with boundary layers over the shock-tube side-wall.

Steady and unsteady 3-D interactive boundary layers

The paper describes theoretical and computational research on 3-D steady and unsteady flows at medium-to-high Reynolds numbers (Re), aimed at increasing understanding of 3-D separation and boundary-layer transition. Concerning steady 3-D flows first, an interactive-boundary-layer (IBL) formulation for 3-D laminar flow of an incompressible fluid over a surface-mounted obstacle is addressed computationally and compared with other methods at various Re. The computational approach is designed deliberately to capture the extra ellipticity present due to the three-dimensionality, making use of skewed shears in linear quasi-planar sweeps of the boundary layer and local updating in the 3-D interaction law. Results including separation are presented for a range of Re and obstacle heights, together with grid-effect studies, and comparisons are made, first with triple-deck predictions for high Re and, second, with an alternative IBL approach presented in a companion work. The latter and the current work together yield a broad agreement on predictions for the 3-D flow, stretching from the triple-deck through the IBL to thin-layer Navier-Stokes predictions, over a wide range of Re. Second, the computational approach is extended to unsteady 3-D flows, for the triple-deck limit including linear and nonlinear Tollmien-Schlichting waves. Results for small and non-small disturbances and comparisons are presented, showing fairly encouraging agreement between theory, computations and experiments.

Simulations of bifurcation processes of reflected-shock waves in ionizing argon in a shock tube

Abstract A finite-difference procedure, which numerically solves the thin-layer Navier-Stokes equations with ionization reactions, has been developed to predict ionizing nonequilibrium flows generated by strong shock waves. By using the procedure, simulations were carried out for bifurcated reflected-shock waves in ionizing argon in a shock tube. Results are shown for the reflection process of an ionizing shock moving at Mach No. 14 in argon at 400 Pa and 298 K. Computed flowfields are compared with snap-shot shadowgraphs for reflected-shock waves in ionizing argon. Qualitative agreement between the simulations and the experiments is excellent.

Unsteady separation over maneuvering bodies

Abstract : Development of a new boundary layer code has reached the status to permit meaningful applications. Computations have been carried out for the initiation of separation in the symmetry plane of a prolate spheroid of slenderness ratio 1/4, impulsively started into forward motion at zero incidence and also at 50 degrees angle. This case serves as validation by comparing with previously published results of Xu and Wang (ref.1), and also demonstrates that our method gives information of flow near the rear stagnation point not available in the literature. More studies have been performed on the optimization of surface suction to delay or prevent the unsteady separation for an impulsively started circular cylinder. Here the methodology should be of interest. Work on an unsteady three-dimensional thin-layer Navier-Stokes code, however, is progressing slowly. Keywords: Unsteady separation, three dimensional moving body, Separation control.

Unsteady inviscid and viscous computations for vortex-dominated flows

The unsteady supersonic flow around a rigid sharp-edged delta wing is solved using the unsteady Euler and thin-layer Navier-Stokes equations. The problem is formulated relative to a moving frame of reference so that computation of the grid motion is not required. A three-dimensional, implicit, aproximately factored, central-differencing, finite-volume scheme is used to obtain the time-accurate, unsteady, locally conical flow. The inviscid and viscous results are compared during the periodic response of a wing undergoing forced rolling-oscillation motion. The results show substantial differences in the surface pressure results and other distributed flow characteristics. These differences are attributed to the inviscid and viscous flow predictions in the vortex-dominated regions. The results of the total loads do not show any significant differences.

Computation and visualization of leading edge vortex flows

The modeling of vortical flows is a continuing requirement for the design and analysis of flight vehicles. In this paper, the computation of leading edge vortices is considered. The solution of the laminar, thin layer Navier-Stokes equations for a transonic delta wing is presented as a representative example. Issues relating to the visualization of the results are discussed, and illustrations using the newly developed software VISUAL3 on a Stardent graphics supercomputer are included.

Simulation of a delta wing with two jets in ground effect

Numerical simulations of the flow about a 60° delta planform wing, equipped with two thrust reverser jets, in ground effect are presented. These simulations are based on the solution of the three-dimensional thin-layer Navier-Stokes equations on a body-conforming, overset, multiple grid topology. An implicit, two-factor, numerical scheme with flux-splitting in the streamwise direction is used. The first set of simulations involves the validation of the numerical scheme by studying the impact of numerical parameters (e.g. grid refinement and dissipation levels) on the solution. The second set of simulations is performed to understand the effect of flow-field parameters (such as the height of the delta wing above the ground plane, the jet size, and the jet exit temperature) on the solution. Flow features such as jet deformation and ground vortex formation observed in experimental flow visualizations are captured. Comparisons with experimental data confirm the ability to simulate the loss of wing-borne lift, commonly referred to as “suck-down,” as the delta planform flies at slow speeds in close proximity to the ground. Detailed analysis of the numerical results has given insight into the structure of the ground vortex and the mechanisms of lift loss.

Simulations of reflected-shock waves in shock tubes taking account of side-wall boundary-layer effects.

Numerical simulations are conducted for behaviors of reflected-shock waves interacting with side-wall boundary-layers in shock tubes. A finite-difference procedure is developed as a solver of the thin-layer Navier-Stokes equations. This procedure, which combines the Lax-Wendroff-FCT scheme with the Crank-Nicolson scheme, is applied to initial and boundary conditions in which effects of shock-tube side-wall boundary-layers. are taken account of by use of the Pohlhausen polynomials. Computations are carried out for reflected-shock waves in nitrogen and in argon gas, respectively. Bifurcated reflected-shock waves, observed experimentally in polyatomic gases in shock tubes, are reproduced in simulations for nitrogen gas and compared with semiempirical theory for bifurcated reflected-shock waves. Good agreements between them lead us to conclude that the present procedure can simulate essential parts of interactions between reflected-shock waves and side-wall boundary-layers.

A computational study of the shock-wave reflection in a shock-tube and the filmwise condensation of a vapour on shock-tube walls.

The present paper deals with the reflection process of a shock wave on a shock-tube endwall in a vapour and with the growth process of liquid films formed both on the end-and sidewalls. Two-dimensional thin-layer Navier-Stokes equations for the vapour and two-dimensional heat conduction equations both for the liquid films and the shock-tube walls are solved under molecular gas dynamical boundary conditions by finite difference methods. It is shown that flow fields in the shock wave reflection region are very complicated because of the interaction of the reflected shock wave with a boundary layer on the sidewall. But there is no large difference between the present results of film growth and that of the one-dimensional model. The results are compared with experimental ones.

Computation of viscous transonic flow over porous airfoils

The viscous effects on transonic flow past an airfoil which contains a shallow cavity beneath a porous surface are studied numerically. The porous region occupies a small portion of the total airfoil surface, and is located near the shock. Both an interactive boundary layer (IBL) algorithm and a thin-layer Navier-Stokes (TLNS) algorithm have been modified for use in studying the outer flow, whereas a stream-function formulation has been used to model the inner flow in the small cavity. The coupling procedure at the porous surface is based on Darcy's law and on the assumption of a constant total presusre in the cavity. In addition, a modified Baldwin-Lomax turbulence model is used to consider the transpired turbulent boundary layer in the TLNS approach, and the Cebeci-Smith turbulence model is used in the IBL approach. According to the present analysis, a porous surface can reduce the wave drag appreciably, but it can also increase viscous losses. As has been observed experimentally, the numerical results indicate that the total drag is reduced at higher Mach numbers and increased at lower Mach numbers when the angles of attack are small. Furthermore, the streamline patterns of passive-shock and boundary-layer interaction are revealed in this study.

Iteration strategy for upwind/relaxation solutions to the thin-layerNavier-Stokes equations

AN efficient iteration strategy for supersonic flow solutions to the and thin-layer Navier-Stokes equations is described. Modifications are made to a previously developed upwind/relaxation algorithm to suppress the upstream influence within the subsonic part of the boundary layer through restriction of the streamwise pressure gradient. A parabolized solution then is efficiently obtained by marching downstream and locally iterating each crossflow plane to convergence. The solution, fully conservative and second-order-accurate, provides an excellent solution for problems without strong adverse streamwise pressure gradients. If necessary, the thin-layer solution may be recovered by further iteration using the original upwind/relaxation algorithm.

Numerical simulation of vortical flows over a strake-delta wing

The vortical flow fields over a double-delta wing with sweep angles 80 and 60 deg are investigated by means of numerical simulations based on the LU-ADI code of Fujii and Obayashi (1986) for the three-dimensional thin-layer Navier-Stokes equations. The results obtained using an 850,000-point grid at angle of attack alpha = 6-40 deg are presented graphically, compared with experimental data, and characterized in detail. Qualitatively accurate simulations are obtained for transition phenomena such as strake/wing-vortex interaction at alpha = 12 deg, bubble-type vortex breakdown near the trailing edge at alpha = 30 deg, and spiral-type breakdown at alpha = 35 deg. It is suggested that quantitative simulations can be achieved at higher grid resolution.

Explicit Runge-Kutta method for unsteady rotor/stator interaction

A quasi-three-dimensional rotor/stator analysis has been developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant timestep. Results are shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Euler and Navier-Stokes results are compared on the scaled single- and multi-passage machine. The method is relatively fast and the quasi-three-dimensional formulation is applicable to a wide range of turbomachinery geometries.

Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-Stator Interaction; Part I1 - Results

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial to efforts to improve current turbomachinery designs, and the computational approach can be used to great advantage in this regard. This study presents a finite-difference, unsteady, thin-layer Navier-Stokes solution to the flow within an axial turbine stage. The computational methodology developed for this simulation is presented in Part I of this paper. The calculation includes end-wall and tip-leakage effects. Results in the form of time-averaged surface pressures, pressure amplitudes (corresponding to the pressure fluctuation in time), near-surface velocity vectors, and pressure contours in the passage areas are presented. The numerical results are compared with experimental data wherever possible and the agreement between the two is found to be good.

Three-dimensional Navier-Stokes simulations of turbine rotor-stator interaction. Part I - Methodology

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial to efforts to improve current turbomachinery designs, and the computational approach can be used to great advantage in this regard. This study presents a finite-difference, unsteady, thin-layer Navier-Stokes approach to calculating the flow within an axial turbine stage. The relative motion between the stator and rotor airfoils is made possible with the use of patched grids that move relative to each other. The calculation includes end-wall and tip-leakage effects. The numerical methodology is presented in detail in the present paper (Part I). The computed results and comparisons of these results with experimental data are presented in a companion paper (Part II).

Simulation of transonic viscous flow over a fighter-like configuration including inlet

The simulation of transonic viscous flow over a modified F-16A including inlet is presented. A zonal approach is utilized which allows appropriate clustering suitable for viscous calculations on all solid surfaces. Computational efficiency is enhanced by solving the thin-layer Navier-Stokes equations in viscous zones adjacent to the aircraft and the Euler equations on those coarse zones away from the aircraft. The flow conditions for this transonic case are freestream Mach number of 0.9, angle of attack of 4.12 deg, and a Reynolds number (based on root chord) of 4.5 million. A total of 19 zones are utilized yielding a total of 350,000 grid points. This case required about 3000 iterations to reduce the residual by three orders, which takes about 10 hr of CPU time on the Cray X-MP/48 computer. Pressure distributions on the wing and on cross sections through the inlet region compare favorably with the experimental data for this transonic case.