Spatial direct numerical simulation of high-speed boundary-layer flows part I: Algorithmic considerations and validation

A highly accurate algorithm for the direct numerical simulation (DNS) of spatially evolving high-speed boundary-layer flows is described in detail and is carefully validated. To represent the evolution of instability waves faithfully, the fully explicit scheme relies on non-dissipative high-order compact-difference and spectral collocation methods. Several physical, mathematical, and practical issues relevant to the simulation of high-speed transitional flows are discussed. In particular, careful attention is paid to the implementation of inflow, outflow, and far-field boundary conditions. Four validation cases are presented, in which comparisons are made between DNS results and results obtained from either compressible linear stability theory or from the parabolized stability equation (PSE) method, the latter of which is valid for nonparallel flows and moderately nonlinear disturbance amplitudes. The first three test cases consider the propagation of two-dimensional second-mode disturbances in Mach 4.5 flat-plate boundary-layer flows. The final test case considers the evolution of a pair of oblique second-mode disturbances in a Mach 6.8 flow along a sharp cone. The agreement between the fundamentally different PSE and DNS approaches is remarkable for the test cases presented.     

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

A study of instabilities in incompressible boundary-layer flow on a flat plate is conducted by spatial direct numerical simulation (DNS) of the Navier-Stokes equations. Here, the DNS results are used to evaluate critically the results obtained using parabolized stability equations (PSE) theory and to study mechanisms associated with breakdown from laminar to turbulent flow. Three test cases are considered: two-dimensional Tollmien-Schlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-flow distortion component of the two-dimensional test problem. This discrepancy is attributed to far-field boundary-condition differences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic flow and a variation of the disturbance frequency result in better agreement with the experiments.     

A new method for computing turbulence in compressible laminar-turbulent transition and boundary layers--PSE＋DNS

A new method for computing laminar-turbulent transition and turbulence in compressible boundary layers is proposed. It is especially useful for computation of laminar-turbulent transition and turbulence starting from small-amplitude disturbances. The laminar stage, up to the beginning of the breakdown in laminar-turbulent transition, is computed by parabolized stability equations (PSE). The direct numerical simulation (DNS) method is used to compute the transition process and turbulent flow, for which the inflow condition is provided by using the disturbances obtained by PSE method up to that stage. In the two test cases incfuding a subsonic and a supersonic boundary layer, the transition locations and the turbulent flow obtained with this method agree well with those obtained by using only DNS method for the whole process. The computational cost of the proposed method is much less than using only DNS method.     

Non-equilibrium turbulent phenomena in transitional flat plate boundary-layer flows

Many recent laboratory experiments and numerical simulations support a non-equilibrium dissipation scaling in decaying turbulence before it reaches an equilibrium state. By analyzing a direct numerical simulation(DNS) database of a transitional boundary-layer flow, we show that the transition region and the non-equilibrium turbulence region, which are located in different streamwise zones, present different non-equilibrium scalings. Moreover, in the wall-normal direction, the viscous sublayer, l...     

Direct numerical simulation of laminar breakdown in high-speed,axisymmetric boundary layers

The laminar breakdown of high-speed, axisymmetric boundary-layer flow is simulated numerically by solving the compressible Navier-Stokes equations using spectral collocation and high-order compact-difference techniques. Numerical test cases include Mach 4.5 flow along a hollow cylinder and Mach 6.8 flow along a sharp cone. From initial states perturbed by second-mode primary and subharmonic (H-type) secondary disturbances, the well-resolved (temporal) calculations proceed well into the laminar breakdown stages, characterized by saturation of the primary and secondary instability waves, explosive growth of higher harmonics, and rapid increase in the wall shear stress. The numerical results qualitatively replicate two previously unexplained phenomena which have been observed in high-speed transition experiments: the appearance of so-called rope-like waves and the precursor transition effect, in which transitional flow appears to originate near the critical layer well upstream of the transition location at the wall. The numerical results further reveal that neither of these effects can be explained, even qualitatively, by linear stability theory alone. Structures of rope-like appearance are shown to arise from secondary instability. Whereas certain features of the precursor transition effect also emerge from secondary instability theory, its nature is revealed to be fundamentally nonlinear.This research was accomplished largely while the first author was in residence as a National Research Council Associate at NASA Langley Research Center, Hampton, VA 23665, USA.     

Hypersonic boundary-layer transition on a flared cone

Transition on a flared cone with zero angle of attack was studied in our newly established Mach 6 quiet wind tunnel (M6QT) via wall pressure measurement and flow visualization. High-frequency pressure transducers were used to measure the second-mode waves’ amplitudes and frequencies. Using pulsed schlieren diagnostic and Rayleigh scattering technique, we got a clear evolution of the second-mode disturbances. The second-mode waves exist for a long distance, which means that the second-mode waves grow linearly in a large region. Strong Mach waves are radiated from the edge of the boundary layer. With further development, the second-mode waves reach their maximum magnitude and harmonics of the second-mode instability appear. Then the disturbances grow nonlinearly. The second modes become weak and merge with each other. Finally, the nonlinear interaction of disturbance leads to a relatively quiet zone, which further breaks down, resulting in the transition of the boundary layer. Our results show that transition is determined by the second mode. The quiet zone before the final breakdown is observed in flow visualization for the first time. Eventual transition requires the presence of a quiet zone generated by nonlinear interactions.     

Direct Numerical Simulation of Hypersonic Boundary-Layer Flow on a Flared Cone

The forced transition of the boundary layer on an axisymmetric flared cone in Mach 6 flow is simulated by the method of spatial direct numerical simulation (DNS). The full effects of the flared afterbody are incorporated into the governing equations and boundary conditions; these effects include nonzero streamwise surface curvature, adverse streamwise pressure gradient, and decreasing boundary-layer edge Mach number. Transition is precipitated by periodic forcing at the computational inflow boundary with perturbations derived from parabolized stability equation (PSE) methodology and based, in part, on frequency spectra available from physical experiments. Significant qualitative differences are shown to exist between the present results and those obtained previously for a cone without afterbody flare. In both cases, the primary instability is of second-mode type; however, frequencies are much higher for the flared cone because of the decrease in boundary-layer thickness in the flared region. Moreover, Goertler modes, which are linearly stable for the straight cone, are unstable in regions of concave body flare. Reynolds stresses, which peak near the critical layer for the straight cone, exhibit peaks close to the wall for the flared cone. The cumulative effect appears to be that transition onset is shifted upstream for the flared cone. However, the length of the transition zone may possibly be greater because of the seemingly more gradual nature of the transition process on the flared cone. Received 20 March 1997 and accepted 21 May 1997     

Direct numerical simulation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer

A numerical algorithm and code are developed and applied to direct numerical simulation (DNS) of unsteady two-dimensional flow fields relevant to stability of the hypersonic boundary layer. An implicit second-order finite-volume technique is used for solving the compressible Navier–Stokes equations. Numerical simulation of disturbances generated by a periodic suction-blowing on a flat plate is performed at free-stream Mach number 6. For small forcing amplitudes, the second-mode growth rates predicted by DNS agree well with the growth rates resulted from the linear stability theory (LST) including nonparallel effects. This shows that numerical method allows for simulation of unstable processes despite its dissipative features. Calculations at large forcing amplitudes illustrate nonlinear dynamics of the disturbance flow field. DNS predicts a nonlinear saturation of fundamental harmonic and rapid growth of higher harmonics. These results are consistent with the experimental data of Stetson and Kimmel obtained on a sharp cone at the free-stream Mach number 8.     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.     

Direct numerical simulation of compressible turbulent flows

This paper reviews the authors＇ recent studies on compressible turbulence by using direct numerical simulation （DNS）,including DNS of isotropic（decaying） turbulence, turbulent mixing-layer,turbulent boundary-layer and shock/boundary-layer interaction.Turbulence statistics, compressibility effects,turbulent kinetic energy budget and coherent structures are studied based on the DNS data.The mechanism of sound source in turbulent flows is also analyzed. It shows that DNS is a powerful tool for the mechanistic study of compressible turbulence.     

Numerical simulation of boundary-layer transition at Mach two

The linear and early nonlinear stages of boundary-layer transition at free-stream Mach numberM ==2.0 are investigated by direct numerical simulation of the compressible Navier-Stokes equations. Results from simulations with a large computational box and small-amplitude random initial conditions are compared with linear stability theory. The growth rates of oblique waves are reproduced correctly. Two-dimensional waves show a growth that is modulated in time, indicating the presence of an extra unstable mode which moves supersonically relative to the free stream. Further simulations are conducted to investigate the nonlinear development of two- and three-dimensional disturbances The transition due to oblique disturbance waves is the most likely cause of transition at this Mach number, and is found to lead to the development of strong streamwise vortices.     

Numerical experiments on stability and transition in plane channel flow

Various secondary and tertiary instabilities in plane channel flow are explored via time-dependent numerical simulations using the incompressible Navier-Stokes equations. Comparisons are made between transitional flows at Reynolds numbers 1500, 5000, and 8000. The lambda vortex, detached shear layer, and inverted vortex regions are identified and the origin of the latter is explained. The laminar breakdown of the Re=1500 flow is computed with high resolution and the nature of its ensuing hairpin eddies is clarified by numerical particle paths. The potential of center-mode rather than wall-mode transitions is proposed and the resulting flow structure is described.     

PSE as applied to problems of transition in compressible boundary layers

A new idea of using the parabolized stability equation (PSE) method to predict laminar-turbulent transition is proposed. It is tested in the prediction of the location of transition for compressible boundary layers on flat plates, and the results are compared with those obtained by direct numerical simulations (DNS). The agreement is satisfactory, and the reason for this is that the PSE method faithfully reproduces the mechanism leading to the breakdown process in laminar-turbulent transition, i. e., the modification of mean flow profile leads to a remarkable change in its stability characteristics.     

A Limiting Solution for Flow Past a Delta Wing in the Presence of Supercritical Flow Regions

The flow past a planar delta wing is studied for strong interaction between the boundary layer and the outer supersonic flow. An analytic investigation is carried out using the Newtonian passage to the limit in which the specific heat ratio tends to unity and the Mach and Reynolds numbers to infinity. Possible flow regimes are classified for various wing aspect ratios. For determining the supercritical-subcritical flow transition line an analytic expression, correct to the second approximation, is obtained for flow past a cold wing with a fairly large aspect ratio in which the transverse boundary-layer flows are insignificant.     

Experimental and numerical study of ?ow structures of the second-mode instability

Flow structures of a Mach 6 transitional boundary layer over a 260 mm long flared cone are investigated by the particle image velocimetry(PIV). Particle images near the curved wall are initially transformed into surface-fitted orthogonal coordinates and spliced with their 180?-symmetric images to satisfy a no-slip condition at the wall.The results are then reversely transformed to the physical domain. Direct numerical simulation(DNS) is also performed to validate the experimental results. The experimental and numerical results are in agreement, indicating a strong dilatation process within the second-mode instability.     

On laminar bouudary layers with suction

In this paper, we obtain the asymptotic solution of the general equation for laminar boundary-layer flows with suction. Formulae for calculating the displacement thickness, momentum thickness, an’d skin friction are then derived. Furthermore, the problem of determining the separation point is dealt with. Finally, as a numerical example, we compute certain characteristic boundary-layer parameters for the case of uniform flow over flat plate with constant suction. Our numerical results obtained are in good agreement with those of Iglisch.     

On the nature of PSE approximation

The recently developed method of parabolized stability equations (PSE) offers a fast and efficient way of analyzing the spatial growth of linear and nonlinear (convective) disturbances in shear layers. For incompressible flows, the governing equations may be represented either in primitive variables or by using other formulations obtained by eliminating the pressure gradient (e.g., vorticity-streamfunction formulation). On the other hand, for compressible flows, primitive variables offer a natural and the only choice. We show that primitive-variable formulation is not well-posed due to the ellipticity introduced by the term and the marching solution eventually blows up for a sufficiently small step size. However, it is shown that this difficulty can be overcome if the minimum step size is greater than the inverse of the real part of the streamwise wave number, _r. An alternative is to drop the term, in which case the residual ellipticity is of no consequence for marching computations with much smaller step sizes. However, the ellipticity cannot be completely removed. Results obtained with streamfunction and vorticity-velocity formulations also show that the numerical difficulties arise for a sufficiently small marching step size. This step-size restriction can be overcome by dropping thed/dx term from the governing equations. The effect of this term on solution accuracy is negligible for Blasius flow but not so for rotating-disk flow.This work was performed under AFOSR contract F49620-91-C-0014.     

Turbulent breaking of overturning gravity waves below a critical level

The interaction of an internal gravity wave with its evolving critical layer and the subsequent generation of turbulence by overturning waves are studied by three-dimensional numerical simulations. The simulation describes the flow of a stably stratified Boussinesq fluid between a bottom wavy surface and a top flat surface, both without friction and adiabatic. The amplitude of the surface wave amounts to about 0.03 of the layer depth. The horizontal flow velocity is negative near the lower surface, positive near the top surface with uniform shear and zero mean value. The bulk Richardson number is one. The flow over the wavy surface induces a standing gravity wave causing a critical layer at mid altitude. After a successful comparison of a two-dimensional version of the model with experimental observations (Thorpe [21]), results obtained with two different models of viscosity are discussed: a direct numerical simulation (DNS) with constant viscosity and a large-eddy simulation (LES) where the subgrid scales are modelled by a stability-dependent first-order closure. Both simulations are similar in the build-up of a primary overturning roll and show the expected early stage of the interaction between wave and critical level. Afterwards, the flows become nonlinear and evolve differently in both cases: the flow structure in the DNS consists of coherent smaller-scale secondary rolls with increasing vertical depth. On the other hand, in the LES the convectively unstable primary roll collapses into three-dimensional turbulence. The results show that convectively overturning regions are always formed but the details of breaking and the resulting structure of the mixed layer depend on the effective Reynolds number of the flow. With sufficient viscous damping, three-dimensional turbulent convective instabilities are more easily suppressed than two-dimensional laminar overturning.     

Studies on nonlinear stability of three-dimensional H-type disturbance

The three-dimensional H-type nonlinear evolution process for the problem of boundary layer stability is studied by using a newly developed method called parabolic stability equations (PSE). The key initial conditions for sub-harmonic disturbances are obtained by means of the secondaryinstability theory. The initial solutions of two-dimensional harmonic waves are expressed in Landau expansions. The numerical techniques developed in this paper, including the higher order spectrum method and the more effective algebraic mapping for dealing with the problem of an infinite region, increase the numerical accuracy and the rate of convergence greatly. With the predictor-corrector approach in the marching procedure, the normalization, which is very important for PSE method, is satisfied and the stability of the numerical calculation can be assured. The effects of different pressure gradients, including the favorable and adverse pressure gradients of the basic flow, on the “H-type“ evolution are studied in detail. The results of the three-dimensional nonlinear “H-type“ evolution are given accurately and show good agreement with the data of the experiment and the results of the DNS from the curves of the amplitude variation, disturbance velocity profile and the evolution of velocity.     

Small-Scale Roughness Effects on Laminar Separation

In this study an interacting boundary-layer (IBL) algorithm is used to investigate small-scale surface roughness effects on the laminar separation mechanism, where small-scale is intended to mean roughness fully contained within the boundary layer. Steady, laminar breakaway separation is computed for two-dimensional flow past a symmetric biconvex airfoil with small-scale roughness elements added to the surface. In this case the flow separation is generated at the trailing edge of a biconvex airfoil, but results are relevant to laminar separation points in general such as that occurring in the leading-edge region. The study is interested primarily in the laminar separation point, and not necessarily the entire bubble and downstream region. The use of the IBL method made it possible to achieve the required fine resolution in areas of interest. For some roughness geometries and flow conditions, up to 15000 grid points (over 4 million total grid points) were used in the streamwise direction to capture the resulting flow physics, which would still be time restrictive with a full Navier–Stokes algorithm. A number of different small-scale roughness configurations were evaluated including variations of roughness height, wavelength, distribution, and geometry. Results from this work show that small-scale roughness can alter the characteristics of the laminar separation point in low-speed flows.