Thin-layer Navier-Stokes Equations Research Articles

Cone-derived waverider flowfield simulation including turbulence and off-design conditions

Using the numerical scheme of Roe's flux-difference splitting, the Baldwin-Lomax algebraic turbulent model, and the time-averaged thin-layer Navier-Stokes equation, the flowfields for a selected Rasmussen's waverider forebody are investigated. Specific cases include turbulent flow on-design condition (M^ = 4, a = 0), turbulent flow off-design condition (Moo = 4, a = 5 deg), and laminar (and Euler) flow off-design condition (M^ = 5, a = 0). For comparison purposes, some results previously reported by the authors are presented. The off-design flowfields are more complicated than the on-design ones. For the M^ = 5 waverider, there is very strong stock-wave/boundarylayer interaction. For the a = 5 deg waverider, there is a detached shock, resulting in a small separation bubble and a vortex on the upper surface. The shock-wave/boundary-layer interaction and shock attachment pattern strongly affect the waverider pressure and local skin-friction coefficient distributions.

Computed effects of heat transfer on the transonic flow over an aerofoil

The effects of the model surface to freestream adiabatic temperature ratio (Tw/Ta^) on subsonic flows at zero pressure gradient and transonic flow over a NACA0012 aerofoil are evaluated using a computational fluid dynamic approach. The analysis, based on the thin layer Navier-Stokes equations with a Bald win-Lorn ax turbulence model, indicated that surface heat transfer has significant effects on both subsonic and transonic flows confirming some of the experimental data available. The results have implications in wind-tunnel testing at nonadiabatic surface conditions.

Grid sensitivity and aerodynamic optimization of generic airfoils

An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B- Splines) for defining the airfoil geometry. An interactive algebraic grid generation technique is employed to generate C-type grids around airfoils. The grid sensitivity of the domain with respect to geometric design parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations such as a wing or fuselage. The aerodynamic sensitivity coefficients are obtained by direct differentiation of the compressible two-dimensional thin-layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the airfoil surface. Results demonstrate a substantially improved design due to maximized lift/drag ratio of the airfoil.

Three-dimensional boundary-layer transition on a swept wing at Mach 3.5

Transition on a swept-wing leading-edge model at Mach 3.5 is investigated. Surface pressure and temperature measurements are obtained in the NASA Langley Research Center Supersonic Low-Disturbance Tunnel. For one case, temperature-sensitive paint and a sublimating chemical are used to visualize surface flow features such as transition location. The experimental data are compared with 1) mean-flow results computed as solutions to the thin-layer Navier-Stokes equations and 2) N-factors obtained using the envelope e N method. The experimental and computational results compare favorably in most cases. In particular, N ≃ 13 correlates best with the observed transition location over a range of freestream unit Reynolds numbers and angles of attack. Computed traveling crossflow disturbances with frequencies of 40-60 kHz have the largest N factors, and the surface flow visualizations reveal smooth transition fronts with only faint evidence of stationary crossflow vortices. These results suggest that transition is probably dominated by traveling, rather than stationary, crossflow disturbances for the present model.

Flow of a fissioning gas in an outflow disk magnetohydrodynamic generator

The influence of fissioning and magnetohydrodynamic (MHD) interaction on the steady, supersonic flow of a compressible, turbulent, weakly ionized, fissioning gas in an outflow disk MHD generator is investigated. The two-dimensional (r, z) MHD flow is modeled using the thin-layer Navier-Stokes equations with MHD and fission power density source terms and Maxwell's equations, simplified by the MHD approximations and by neglecting the induced magnetic field. Simple plasma physics transport property models are developed for the collisiondominated, weakly ionized plasma in which fission fragment induced ionization provides the dominant source of conduction electrons. The two-dimensional MHD solution methodology is used to characterize the effects of fission power density and variable applied magnetic induction levels on the spatial profiles of important generator variables. A comparison of the predictions of the two-dimensional MHD solver with those of a quasi-onedimensional Euler solver with MHD and fission source terms is used to discuss the relative influence of twodimensional effects on the generator flowfield and on performance levels. The low MHD interaction and generator performance levels predicted for the supersonic fissioning plasma suggest that the fission fragment induced ionization, by itself, is insufficient to effect the electrical conductivity levels needed for practical power generation in the outflow disk MHD generator configuration of this study.

Unsteady Aerodynamic Characteristics of Pitch-Oscillating Transonic Cascade Case with an In-Passage Shock.

A numerical study based on the thin-layer Navier-Stokes equations was carried out on the pitch-oscillating two-dimensional transonic cascade around the midchord. A Baldwin-Lomax turbulence model was used to simulate the effects of turbulence. The unsteady aerodynamic characteristics and the role of shock wave behavior were investigated. The numerical results indicate that the shock wave movement acts as an aerodynamically unstable factor at high reduced frequency and in a range of interblade phase angle, but the pitch-oscillating transonic cascade remains aerodynamically stable in the range of calculations. However it becomes unstable if the pitching axis is located in front of X=0. 37.

An Approximately Factored Incremental Strategy for Calculating Consistent Discrete Aerodynamic Sensitivity Derivatives

In this study involving advanced fluid flow codes, an incremental iterative formulation (also known as the "delta" or "correction" form), together with the well-known spatially split approximate factorization algorithm, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For smaller 2D problems a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods are needed for larger 2D and future 3D applications, however, because direct methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to a lack of diagonal dominance and perhaps ill-conditioning of the coefficient matrix; this problem can be overcome when these equations are cast in the incremental form. These and other benefits are discussed herein. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two sample airfoil problems: (1) subsonic low Reynolds number laminar flow; and (2) transonic high Reynolds number turbulent flow.

Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades

A three-dimensional nonlinear time-marching method of solving the thin-layer Navier–Stokes equations in a simplified form has been developed for blade flutter calculations. The discretization of the equations is made using the cell-vertex finite volume scheme in space and the four-stage Runge–Kutta scheme in time. Calculations are carried out in a single-blade-passage domain and the phase-shifted periodic condition is implemented by using the shape correction method. The three-dimensional unsteady Euler solution is obtained at conditions of zero viscosity, and is validated against a well-established three-dimensional semi-analytical method. For viscous solutions, the time-step limitation on the explicit temporal discretization scheme is effectively relaxed by using a time-consistent two-grid time-marching technique. A transonic rotor blade passage flow (with tip-leakage) is calculated using the present three-dimensional unsteady viscous solution method. Calculated steady flow results agree well with the corresponding experiment and with other calculations. Calculated unsteady loadings due to oscillations of the rotor blades reveal some notable three-dimensional viscous flow features. The feasibility of solving the simplified thin-layer Navier–Stokes solver for oscillating blade flows at practical conditions is demonstrated.

Vortex-wing interaction of a close-coupled canard configuration

The thin-layer Navier-Stokes equations are solved numerically to investigate the effects of canard vertical position on a close-coupled, canard-wing-body configuration at a transonic Mach number of 0.90 and angles of attack ranging from — 2 to 12 deg. Canard-wing interactions are investigated for the canard positioned above, coplanar with, and below the wing (high-, mid- and low-canard positions, respectively). The computational results show favorable canard-wing interactions for the high- and midcanard configurations. The unfavorable lift and drag characteristics for the low-canard configuration are examined by analyses of the low-canard flowfield structure and found to be directly attributed to the interaction between the canard vortex and the wing surface. At relatively low angles of attack, the low-canard vortex passes under the wing surface and induces low pressures on the wing lower surface. As angle of attack is increased, the low-canard vortex impacts the wing surface and is split into two distinct vortices.

Reduction of blade-vortex interaction noise through porous leading edge

The effect of the porous leading edge of an airfoil on the blade-vortex interaction noise, which dominates the far-field acoustic spectrum of the helicopter, is investigated. The thin-layer Navier-Stokes equations are solved with a high-order upwind-biased scheme and a multizonal grid system. The Baldwin-Lomax turbulence model is modified for considering transpiration on the surface. The amplitudes of the propagating acoustic wave in the near field are calculated directly from the computation. The porosity effect on the surface is modeled in two ways: (1) imposition of prescribed transpiration velocity distribution and (2) calculation of transpiration velocity distribution by Darcy's law. Results show leading-edge transpiration can suppress pressure fluctuations at the leading edge during blade-vortex interaction and consequently reduce the amplitude of propagating noise by 30% at a maximum in the near field.

Compressible Navier-Stokes computations of multielement airfoil flows using multiblock grids

Numerical solutions of the compressible thin-layer Navier-Stokes equations are presented for high-lift multielement airfoil configurations. Multiblock grids are used, allowing straightforward implementation of an approximately factored implicit algorithm. Interfaces between blocks are treated by overlapping the grids and taking one layer of points from neighboring blocks. Turbulence is modeled using the Baldwin-Earth one-equation turbulence model. High-lift applications presented for comparison with wind tunnel data include: an NACA 4412 airfoil with NACA 4415 flap, a GA(W)-1 airfoil with a 29% chord flap at 30-deg flap angle and two gap settings, and a GA(W)-1 airfoil with 15% chord slat and 29% chord flap. Good agreement with experimental data is obtained for cases with fully attached flow or small regions of separated flow. For cases with extensive regions of flow separation, the thickness and extent of the separated regions are underpredicted.

Hypersonic flow in chemical non-equilibrium: A comparison of thin-layer Navier-Stokes and coupled Euler/boundary layer computations

A comparison of the computed flowfields for viscous hypersonic flow in chemical non-equilibrium is presented. Two different approximations of the aerothermochemistry equations have been considered: the thin-layer Navier-Stokes equations and a second-order coupling of the Euler and boundary layer equations. To avoid discrepancies arising from the choice of the physical and chemical models, the test case considered has been defined in a precise manner. In addition, close attention is paid to the appropriate flowfield properties to be compared. The detailed agreement obtained in the computed solutions for hypersonic flow over a hyperboloid provides a verification of the accuracy of the numerical methods employed.

Effect of canard deflection on close-coupled canard-wing-body aerodynamics

The thin-layer Navier-Stokes equations are solved for the flow about a canard-wing-body configuration at transonic Mach numbers of 0.85 and 0.90, angles of attack from -4 to 10 degrees and canard deflection angles from -10 to +10 degrees. Effects of canard deflection on aerodynamic performance, including canard-wing vortex interaction, are investigated. Comparisons with experimental measurements of surface pressures, lift, drag and pitching moments are made to verify the accuracy of the computations. The results of the study show that the deflected canard downwash not only influences the formation of the wing leading-edge vortex, but can cause the formation of an unfavorable vortex on the wing lower surface as well.

Computational study of blowing on delta wings at high alpha

In this study, the flowfield produced by tangential leading-edge blowing on a rounded leading-edge 60-deg delta wing at high angle of attack is investigated computationally by solving the thin-layer Navier-Stokes equations. Steady-state flowHelds are calculated for various angles of attack, with and without the presence of tangential leading-edge blowing. The numerical grid is generated using algebraic grid generation and various interpolation and blending techniques. The jet emanates from a slot with linearly varying thickness, and is introduced into the flowfield using the concept of an actuator plane, thereby not requiring resolution of the jet slot geometry. The Baldwin-Lomax algebraic turbulence model is used to provide turbulent closure. The computational results are compared with those of experiments. The effectiveness of blowing as a rolling moment control mechanism to extend the envelope of controllabili ty is illustrated at different angles of attack. The saturation effect of increased blowing is captured well in the computations. Control reversal noted in similar experimental studies is also observed in the computations.

A computational study of annular cascade flows using aq-ω turbulence model with a wall function

A computational code has been developed for steady viscous flows in three dimensional annular cascades. This code solves a special form of the thin-layer Navier-Stokes equations with a two-equationq-ω turbulence model in curvilinear coordinates using a time asymptotic method for steady state solutions. It employs a scalar implicit approximate factorization in time and a finite volume formulation with second-order upwind-differencing in space. A wall function treatment is implemented at solid boundaries for turbulence equations instead of integration to the wall to relieve gridding requirements. In order to validate the effectiveness of this code, computational studies have been made to access modeling capability for complex turbulent flow fields in three dimensional annular cascade geometries which typically include laminar-turbulent boundary layer transition. The results have been compared with both the computational studies with integration to the wall and the experimental studies. The wall function treatment was found to be reliable by predicting secondary flows and loss contours reasonably well.

PARALLELIZATION OF A CLASS OF IMPLICIT FINITE DIFFERENCE SCHEMES IN COMPUTATIONAL FLUID DYNAMICS

Implicit finite difference schemes are often the preferred numerical methods in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme, however, involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such methods, therefore, are more difficult and can be non-intuitive. Moreover, in practical applications, some sections of the computations are explicit while other sections are implicit in nature. Developing techniques that extract maximum parallelism simultaneously from all sections of an application is considerably harder than extracting parallelism from just an implicit section or an explicit section of the computations. In this paper, we consider issues in the parallelization of techniques that are used for solving Euler and thin-layer Navier-Stokes equations and that require, as a part of the computation, solving large linear systems in the form of block tri-diagonal and/or scalar penta-diagonal matrices. We focus our attention on three-dimensional cases and present schemes that minimize the total execution time when implemented on a message passing system. We describe various partitioning and scheduling strategies for alleviating the effects of the global data dependencies. Analyses of the computation, the communication, and the memory requirements of these methods are presented. The ARC-3D code, developed at NASA Ames, is used as an example application. Performance of the proposed methods is verified on the Victor multiprocessor system, a message passing architecture developed at the IBM T.J. Watson Research Center.

Airfoil Shape Optimization Using Sensitivity Analysis on Viscous Flow Equations

An aerodynamic shape optimization method has previously been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs. This method has also included a flowfield extrapolation (or flow prediction) method based on the Taylor series expansion of an existing CFD solution. The present paper reports on the extension of this method to the thin-layer Navier-Stokes equations in order to account for the viscous effects. Also, to test the method under highly nonlinear conditions, it has been applied to the transonic flows. Initially, the success of the flow prediction method is tested. Then, the overall method is demonstrated by optimizing the shapes of two supercritical transonic airfoils at zero angle of attack. The first one is shape optimized to achieve a minimum drag while obtaining a lift above a specified value. Whereas, the second one is shape optimized for a maximum lift while attaining a drag below a specified value. The results of these two cases indicate that the present method can produce successfully optimized aerodynamic shapes.

Numerical Simulations for Shock-Tube Experiments of Reflected-Shock Waves in Combustible Gas.

The present paper treats numerical simulations for detonation initiations behind reflected-shock waves in a shock tube. The two-dimensional, thin-layer Navier-Stokes equations with chemical effects are numerically solved by use of a method combining the Richitmyer-FCT scheme, the Crank-Nicolson scheme and a chemical calculation step. Effects of chemical reactions are estimated in the simulations by using a simplified reaction model in which rate coefficients are determined so that the induction times and the characteristic times of the exothermic reactions are fitted to data of a H2/O2/Ar mixture. Simulations are carried out, referring to experiments by several authors. Results of simulations reveal a mechanism of triple-shock generation in reaction shock waves. Computed flowfields for strong ignition in hydrogen and oxygen are in good qualitative agreement with visualized ones in the experiments. A simulation referring to mild ignition predicts such a feature that the ignition starts from distinct kernels. It is also predicted that an ignition occurs immediately behind a normal reflected-shock wave, but does not behind a bifurcated oblique shock wave.

NUMERICAL STUDY OF BASE BLEED EFFECTS ON AERODYNAMIC DRAG FOR A TRANSONIC PROJECTILE

ABSTRACT A numerical study is made to analyze the performance of a secant-ogive-cylinder projectile in the transonic regime in terms of aerodynamic drag. At transonic speeds, the base drag contributes a major portion of the total aerodynamic drag, and hence affects projectile's performances significantly. The base bleed method is applied to reduce the base drag by varying the value of parameters, the bleed quantity (I) and the bleed area ratio (ϖ). The implicit, diagonalized, symmetric Total Variation Diminishing (TVD) scheme, accompanied by a suitable grid, is employed to solve the thin-layer axisymmetric Navier-Stokes equations coupled with the Baldwin-Lomax turbulence model. The computed results show that, in comparison with the case without base bleed, an increase in bleed quantity or a higher injection speed due to a smaller bleed area ratio at fixed bleed quantity can result in a base (and total) drag reduction. At Mach number 0.96, the reductions in base drag and total drag can be as high as 64% an...

NUMERICAL SIMULATION OF THREE-DIMENSIONAL SHOCK-SHOCK INTERACTIONS ON A BLUNT BODY

SUMMARY A numerical study is conducted to simulate the effects of extraneous shock impingement on a blunt body in viscous hypersonic flow. The interaction of extraneous shock with the leading-edge shock results in a very complex flow field that contains local regions of high pressure and intense heating. The heating and pressure can be orders of magnitude higher than the peak values in the absence of shock impingement. The flow field is calculated by solving thin-layer Navier-Stokes equations with a finite-volume flux splitting technique developed by van Leer. For a zero or small sweep of the body, a type IV interaction occurs, which produces a lambda shock structure with a supersonic jet embedded in the otherwise subsonic flow; for a moderate sweep of about 25°, a type V interaction occurs in which a subsonic shear layer sandwiched in supersonic flow is produced with a transmitted shock. In the present study, both type IV and type V interactions are investigated. Results of the present numerical investig...