A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers

The capabilities of the triple-deck theory of receptivity for subsonic compressible boundary layers have been thoroughly investigated through comparisons with numerical simulations of the compressible Navier–Stokes equations. The analysis focused on the two Tollmien–Schlichting wave linear receptivity problems arising due to the interaction between a low-amplitude acoustic wave and a small isolated roughness element, and the low-amplitude time-periodic vibrations of a ribbon placed on the wall of a flat plate. A parametric study was carried out to look at the effects of roughness element and vibrating ribbon longitudinal dimensions, Reynolds number, Mach number and Tollmien–Schlichting wave frequency. The flat plate is considered isothermal, with a temperature equal to the laminar adiabatic-wall temperature. Numerical simulations of the full and the linearised compressible Navier–Stokes equations have been carried out using high-order finite differences to obtain, respectively, the steady basic flows and the unsteady disturbance fields for the different flow configurations analysed. The results show that the asymptotic theory and the Navier–Stokes simulations are in good agreement. The initial Tollmien–Schlichting wave amplitudes and, in particular, the trends indicated by the theory across the whole parameter space are in excellent agreement with the numerical results. An important finding of the present study is that the behaviour of the theoretical solutions obtained for $\mathit{Re}\rightarrow \infty$ holds at finite Reynolds numbers and the only conditions needed for the theoretical predictions to be accurate are that the receptivity process be linear and the free-stream Mach number be subsonic.

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ON NONPARALLEL STABILITY OF THREE-DIMENSIONAL COMPRESSIBLE BOUNDARY LAYERS

Modern Physics Letters B ◽

10.1142/s0217984905009766 ◽

2005 ◽

Vol 19 (28n29) ◽

pp. 1503-1506

Author(s):

JIXUE LIU ◽

DENGBIN TANG ◽

GUOXING ZHU

Keyword(s):

Boundary Layers ◽

Stokes Equations ◽

Three Dimensional ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Local Method ◽

Curvature Variation ◽

Compressible Boundary Layers ◽

Curvilinear Coordinate ◽

The Difference

Nonparallel stability of the compressible boundary layers for three-dimensional configurations having large curvature variation on the surface is investigated by using the parabolic stability equations, which are derived from the Navier-Stokes equations in the curvilinear coordinate system. The difference schemes with fourth-order accuracy can be used in the entire computational regions. The global method is combined with the local method using a new iterative formula, thus more precise eigenvalues are obtained, and fast convergences are achieved. Computed curves of the amplification factor and shape functions of disturbances show clearly variable process of the flow stability, and agree well with other available results.

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Direct simulation of turbulent supersonic boundary layers by an extended temporal approach

Journal of Fluid Mechanics ◽

10.1017/s0022112000002718 ◽

2001 ◽

Vol 429 ◽

pp. 187-216 ◽

Cited By ~ 116

Author(s):

THIERRY MAEDER ◽

NIKOLAUS A. ADAMS ◽

LEONHARD KLEISER

Keyword(s):

Numerical Simulation ◽

Mach Number ◽

Direct Numerical Simulation ◽

Boundary Layers ◽

Stokes Equations ◽

Navier Stokes ◽

Mean Flow ◽

Navier Stokes Equations ◽

Number Range ◽

Mach Numbers

The present paper addresses the direct numerical simulation of turbulent zero-pressure-gradient boundary layers on a flat plate at Mach numbers 3, 4.5 and 6 with momentum-thickness Reynolds numbers of about 3000. Simulations are performed with an extended temporal direct numerical simulation (ETDNS) method. Assuming that the slow streamwise variation of the mean boundary layer is governed by parabolized Navier–Stokes equations, the equations solved locally in time with a temporal DNS are modified by a distributed forcing term so that the parabolized Navier–Stokes equations are recovered for the spatial average. The correct mean flow is obtained without a priori knowledge, the streamwise mean-flow evolution being approximated from its upstream history. ETDNS reduces the computational effort by up to two orders of magnitude compared to a fully spatial simulation.We present results for a constant wall temperature Tw chosen to be equal to its laminar adiabatic value, which is about 2.5 T∞, 4.4 T∞ and 7 T∞, respectively, where T∞ is the free-stream temperature for the three Mach numbers considered. The simulations are initialized with transition-simulation data or with re-scaled turbulent data at different parameters. We find that the ETDNS results closely match experimental mean-flow data. The van Driest transformed velocity profiles follow the incompressible law of the wall with small logarithmic regions.Of particular interest is the significance of compressibility effects in a Mach number range around the limit of M∞ ≃ 5, up to which Morkovin's hypothesis is believed to be valid. The results show that pressure dilatation and dilatational dissipation correlations are small throughout the considered Mach number range. On the other hand, correlations derived from Morkovin's hypothesis are not necessarily valid, as is shown for the strong Reynolds analogy.

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A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

Journal of Fluid Mechanics ◽

10.1017/s0022112089003113 ◽

1989 ◽

Vol 209 ◽

pp. 285-308 ◽

Cited By ~ 35

Author(s):

R. J. Bodonyi ◽

W. J. C. Welch ◽

P. W. Duck ◽

M. Tadjfar

Keyword(s):

Numerical Solutions ◽

Free Stream ◽

Stokes Equations ◽

Numerical Study ◽

Navier Stokes ◽

Surface Geometry ◽

Navier Stokes Equations ◽

S Waves ◽

Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

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An implicit marching method for the two-dimensional reduced Navier-Stokes equations at arbitrary mach number

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/0045-7825(78)90065-8 ◽

1978 ◽

Vol 13 (3) ◽

pp. 307-334 ◽

Cited By ~ 7

Author(s):

J.P. Kreskovsky ◽

S.J. Shamroth

Keyword(s):

Mach Number ◽

Stokes Equations ◽

Navier Stokes ◽

Two Dimensional ◽

Navier Stokes Equations ◽

Marching Method

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Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz072 ◽

2020 ◽

Author(s):

David Maltese ◽

Antonín Novotný

Keyword(s):

Mach Number ◽

Initial Data ◽

Error Estimate ◽

Strong Solution ◽

Stokes Equations ◽

Main Tool ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Low Mach Number ◽

Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

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