Compressible Boundary Layer Flow Research Articles

Adjoint-based optimisation of time- and span-periodic flow fields with Space–Time Spectral Method: Application to non-linear instabilities in compressible boundary layer flows

We aim at computing time- and span-periodic flow fields in span-invariant configurations. The streamwise and cross-stream derivatives are discretised with finite volumes while time and the span-direction are handled with pseudo-spectral Fourier-collocation methods. Doing so, we extend the classical Time Spectral Method (TSM) to a Space–Time Spectral Method (S-TSM), by considering non-linear interactions of a finite number of time and span harmonics. For optimisation, we introduce an adjoint-based framework that allows efficient computation of the gradient of any cost functional with respect to a large-dimensional control parameter. Both theoretical and numerical aspects of the methodology are described: evaluation of matrix–vector products with S-TSM Jacobian (or its transpose) by algorithmic differentiation, solution of fixed-points with quasi-Newton method and de-aliasing in time and space, solution of direct and adjoint linear systems by iterative algorithms with a block-circulant preconditioner, performance assessment in CPU time and memory. We illustrate the methodology on the case of 3D instabilities (first Mack mode) triggered within a developing adiabatic boundary layer at M = 4.5. A gradient-ascent method allows to identify a finite-amplitude 3D forcing that triggers a non-linear response exhibiting the strongest time- and span-averaged drag on the flat-plate. In view of flow control, a gradient-descent method finally determines a finite amplitude 2D wall-heat flux that minimises the averaged drag of the plate in presence of the previously determined non-linear optimal forcing.

Estimating Velocity Correlations Using Single Pixel PIV

In this work, we analyze the possibility of determining correlations of turbulent velocity fluctuations from PIV data with improved spatial resolution. The analysis is based on the method presented by Oldenziel et al. (2023), where a joint probability density function of the velocity at two locations is calculated from the single-pixel ensemble correlation for these locations. Synthetic PIV images are used to discuss the occurrence of random and systematic errors and to derive the number of image pairs required for a determination of the correlation coefficient with desired uncertainty, depending on the particle image density. Compared to classical window correlation, the presented method can improve the spatial resolution by about one order of magnitude if a very large number of PIV image pairs are available and/or many statistically independent points with identical flow properties can be used. As an example of application, we show the spatial distribution of a two-point correlation in a compressible turbulent boundary layer flow for a reference point near the wall.

Stable, entropy-pressure compatible subsonic Riemann boundary condition for embedded DG compressible flow simulations

One approach to reducing the computational cost of simulating transitional compressible boundary layer flow is to adopt a near body reduced domain with boundary conditions enforced to be compatible with a computationally cheaper three-dimensional RANS simulation. In such an approach it is desirable to enforce a consistent pressure distribution which is not typically the case when using the standard Riemann inflow boundary conditions. We therefore revisit the Riemann problem adopted in many DG based high fidelity formulations.Through analysis of the one-dimensional linearized Euler equations it is demonstrated that maintaining entropy compatibility with the RANS simulation is important for a stable solution. It is also necessary to maintain the invariant for the Riemann outflow boundary condition in a subsonic flow leaving one condition that can be imposed at the inflow boundary. Therefore the entropy-pressure enforcement is the only stable boundary condition to enforce a known pressure distribution. We further demonstrate that all the entropy compatible inflow Riemann boundary conditions are stable providing the invariant compatible Riemann outflow boundary condition is also respected.Although the entropy-pressure compatible Riemann inflow boundary condition is stable from the one-dimensional analysis, two-dimensional tests highlight divergence in the inviscid problem and neutrally stable wiggles in the velocity fields in viscous simulations around the stagnation point. A two-dimensional analysis about a non-uniform baseflow assumption provides insight into this stability issue (ill-posedness) and motivates the use of a mix of inflow boundary conditions in this region of the flow.As a validation we apply the proposed boundary conditions to a reduced domain of a wing section normal to the leading-edge of the CRM-NLF model taken out of a full three-dimensional RANS simulation at Mach 0.86 and a Reynolds number of 8.5 million. The results show that the entropy-pressure compatible Riemann inflow boundary condition leads to a good agreement in pressure distribution.

Over-reflection of acoustic waves by supersonic exponential boundary layer flows

The two-dimensional acoustic wave equation for inviscid compressible boundary layer flows, i.e. the Pridmore-Brown equation with an exponential velocity profile for homentropic flows, is studied for the reflection and over-reflection of acoustic waves based on the exact solution in terms of the confluent Heun function. The reflection coefficient $R$ , which is the ratio of the amplitude of the reflected to that of the incoming acoustic wave, is determined as a function of the streamwise wavenumber $\alpha$ , the Mach number $M$ and the incident angle $\phi$ of the acoustic waves. Over-reflection refers to $R>1$ , i.e. the reflected wave has a larger amplitude than the incident wave. We prove that, in the supersonic context, energy is always transferred from the base flow to the reflected wave, i.e. $R<1$ does not exist. Meanwhile, this fact is intimately linked to the critical layer. We show that the presence of the critical layer leads to an optimal energy exchange from the base flow into the acoustic wave, i.e. the critical layer ensures $R>1$ . In our analysis, we observe a special phenomenon, resonant over-reflection, which is proven to be closely related to resonant frequencies $\omega _r$ of unstable modes of the temporal stability of the base flow. At resonant frequencies of the first unstable mode, the over-reflection coefficient exhibits an unusual peak in an extremely narrow frequency interval. The maximum values of these peaks are largely synchronized with the variation of the growth rate $\omega _i$ of the unstable modes. In addition, resonant over-reflection appears also at resonant frequencies of other higher unstable modes, but their peaks of the over-reflection coefficient are always smaller than that induced by the first unstable mode.

Towards Density Measurements In A Supersonic Turbulent Boundary Layer

Quantitative mean density measurements using calibrated schlieren technique are demonstrated in a Mach 2 compressible turbulent boundary layer. A calibration procedure, where schlieren and particle image velocimetry measurements are conducted on a symmetric double wedge model, is utilised. By assuming that the shock and expansion fan generated by the double wedge model can be idealised, the density gradient field is calculated based on the measured velocity field. Thus, based on the calibration data, a linear relation from schlieren intensity to density gradient is constructed. Moreover, to resolve a large range of density gradients expected within the boundary layer, schlieren measurements with five different knife edge positions are performed and the results are stitched together. The combined density gradient and the integrated density profiles showed good agreement with the literature, and therefore the current technique may provide a promising avenue to measure the mean density fields in compressible boundary layer flows.

Stability analyses of compressible flat plate boundary layer flow over a mechanically compliant wall

Stability analyses of compressible flat plate boundary layer flow over a mechanically compliant wall

Free-stream coherent structures in parallel compressible boundary-layer flows at subsonic and moderate supersonic Mach numbers

As a first step towards the description of coherent structures in compressible shear flows, we present an asymptotic description of nonlinear travelling-wave solutions of the Navier–Stokes equations in the compressible asymptotic suction boundary layer (ASBL). We consider free-stream Mach numbers $M_\infty$ in the subsonic and moderate supersonic regime so that $0\leqslant M_\infty \leqslant 2$ . We extend the large-Reynolds-number asymptotic theory of Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625) describing ‘free-stream’ coherent structures in incompressible ASBL flow to describe a nonlinear interaction in a thin layer situated just below the free stream. Crucially, the nonlinear interaction equations for the velocity field in this layer are identical to those obtained in the incompressible problem, and thus the asymptotic analysis supporting free-stream coherent structures in compressible ASBL is easily deduced from its incompressible counterpart. The nonlinear interaction produces streaky disturbances to both the velocity and temperature fields, which can grow exponentially towards the wall. We complete the description of the growth of the velocity and thermal streaks throughout the flow by solving the compressible boundary-region equations numerically. We show that the velocity and thermal streaks obtain their maximum amplitude in the unperturbed boundary layer. Increasing the free-stream Mach number enhances the thermal streaks and suppresses the velocity streaks, whereas varying the Prandtl number suppresses the velocity streaks, and can either enhance or suppress the thermal streaks depending on whether the flow is in the subsonic or moderate supersonic regime. Such nonlinear equilibrium states have been implicated in shear transition in incompressible flows; therefore, our results indicate that a similar mechanism may also be present in compressible flows.

Cancellation of Tollmien\u2013Schlichting waves with surface heating

Two-dimensional boundary layer flows in quiet disturbance environments are known to become unstable to Tollmien–Schlichting waves. The experimental work of Liepmann et al. (J Fluid Mech 118:187–200, 1982), Liepmann and Nosenchuck (J Fluid Mech 118:201–204, 1982) showed how it is possible to control and reduce unstable Tollmien–Schlichting wave amplitudes using unsteady surface heating. We consider the problem of an oncoming planar compressible subsonic boundary layer flow with a three-dimensional vibrator mounted on a flat plate, and with surface heating present. It is shown using asymptotic methods based on triple-deck theory that it is possible to choose an unsteady surface heating distribution to cancel out the response due to the vibrator. An approximation based on the exact formula is used successfully in numerical computations to confirm the findings. The results presented here are a generalisation of the analogous results for the two-dimensional problem in Brennan et al. (J Fluid Mech 909:A16-1, 2020).

Large-scale coherent structures in compressible turbulent boundary layers

Abstract

Cost-effective and high-fidelity method for turbulent transition in compressible boundary layer

An efficient high-fidelity simulation approach is investigated for laminar-to-turbulent transition in compressible boundary layer flow. This approach combines large-eddy simulation (LES) with the parabolized stability equations (PSE) analysis. LES is chosen for high-fidelity simulation of the transitional flow, whereas the PSE analysis is used for an efficient treatment of instability modes in the flow. Instability modes from the PSE analysis are assigned as forcing at the inlet of the LES computational domain. A framework of LES combined with PSE (PSE+LES) for the turbulent transition is demonstrated in supersonic boundary layer flow at Mach 3. Complete transition to turbulence via oblique-mode breakdown is well resolved in the current study. Detailed flow features associated with the turbulent transition are compared well with previous direct-numerical simulation (DNS) studies, including the growth of instability modes in the pre-transition regime, the transition range shown in the evolution of the skin friction, and the turbulent boundary layer. It is shown that the current PSE+LES approach can reduce the computational cost by two orders of magnitude, compared to the relevant DNS cost. The current study for supersonic flow and the authors' previous study for subsonic flow imply that the PSE+LES approach can be used in boundary layer flows at various speeds.

Dense-gas effects on compressible boundary-layer stability

A study of dense-gas effects on the stability of compressible boundary-layer flows is conducted. From the laminar similarity solution, the temperature variations are small due to the high specific heat of dense gases, leading to velocity profiles close to the incompressible ones. Concurrently, the complex thermodynamic properties of dense gases can lead to unconventional compressibility effects. In the subsonic regime, the Tollmien–Schlichting viscous mode is attenuated by compressibility effects and becomes preferentially skewed in line with the results based on the ideal-gas assumption. However, the absence of a generalized inflection point precludes the sustainability of the first mode by inviscid mechanisms. On the contrary, the viscous mode can be completely stable at supersonic speeds. At very high speeds, we have found instances of radiating supersonic instabilities with substantial amplification rates, i.e. waves that travel supersonically relative to the free-stream velocity. This acoustic mode has qualitatively similar features for various thermodynamic conditions and for different working fluids. This shows that the leading parameters governing the boundary-layer behaviour for the dense gas are the constant-pressure specific heat and, to a minor extent, the density-dependent viscosity. A satisfactory scaling of the mode characteristics is found to be proportional to the height of the layer near the wall that acts as a waveguide where acoustic waves may become trapped. This means that the supersonic mode has the same nature as Mack’s modes, even if its frequency for maximal amplification is greater. Direct numerical simulation accurately reproduces the development of the supersonic mode and emphasizes the radiation of the instability waves.

Reattachment streaks in hypersonic compression ramp flow: an input–output analysis

We employ global input–output analysis to quantify amplification of exogenous disturbances in compressible boundary layer flows. Using the spatial structure of the dominant response to time-periodic inputs, we explain the origin of steady reattachment streaks in a hypersonic flow over a compression ramp. Our analysis of the laminar shock–boundary layer interaction reveals that the streaks arise from a preferential amplification of upstream counter-rotating vortical perturbations with a specific spanwise wavelength. These streaks are associated with heat-flux striations at the wall near flow reattachment and they can trigger transition to turbulence. The streak wavelength predicted by our analysis compares favourably with observations from two different hypersonic compression ramp experiments. Furthermore, our analysis of inviscid transport equations demonstrates that base-flow deceleration contributes to the amplification of streamwise velocity and that the baroclinic effects are responsible for the production of streamwise vorticity. Finally, the appearance of the temperature streaks near reattachment is triggered by the growth of streamwise velocity and streamwise vorticity perturbations as well as by the amplification of upstream temperature perturbations by the reattachment shock.

Boundary-Layer Characteristics of Compressible Flow past a Heated Cylinder with Viscous Dissipation

We investigate the flow and heat transfer characteristics of compressible boundary-layer flow past a heated cylinder are investigated, taking into account the effect of viscous dissipation. The governing equations of the compressible boundary layer are reduced to a convenient form for integration introducing suitable dimensionless variables. The resulting equations are then solved numerically using the finite difference method for all time, the perturbation method for short time, and the asymptotic method for long time. For different values of the Mach number and the surface temperature parameter , the numerical solutions are demonstrated to show how the local shear stress and the rate of heat transfer at the surface of the cylinder are affected by these parameters. The influences of those parameters on the streamlines and isotherms are also presented here.

Direct numerical simulation of a compressible boundary-layer flow past an isolated three-dimensional hump in a high-speed subsonic regime

This work was supported by the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under Grant No. EP/I037946 and computational resources were also provided through the UK Turbulence resource under EPSRC Grant No. EP/L000261/1. The authors would like to acknowledge the use of Imperial College High Performance Computing facility. S.J.S. would also like to acknowledge support under the RAEng Fellowship through Grant No. 10145/86.

A stability analysis of the compressible boundary layer flow over indented surfaces

This contribution presents a stability analysis for compressible boundary layer flows over indented surfaces. Specifically, the effects of increasing depth D/δ* and Ma∞ number on perturbation time-decay rates and spatial amplification factors are quantified and compared with those of an unindented configuration.The indented surfaces represent aeronautical lifting surfaces endowed with the smooth gap resulting when a filler material applied at the junction of leading-edge and wing-box components retracts upon its curing process. Since the configuration considered is such that the parallel/weakly-parallel assumptions are necessarily compromised, a global temporal stability analysis is considered in this study. Our analysis does not require a parallel flow constrain, and hence it is believed to be valid when two dimensional effects are relevant.We find that small surface modifications enhance certain flow instabilities. An increase in Ma∞ enhances further this behaviour: for the D/δ*=1.5,Ma∞=0.5 case, amplification factors at a given location can be up to 20 times larger than those corresponding to the unindented case.

Thermal Radiation Effect on the MHD Turbulent Compressible Boundary Layer Flow with Adverse Pressure Gradient, Heat Transfer and Local Suction

The combined effect of magnetic field, thermal radiation and local suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient is numerically studied. The magnetic field is constant and applied transversely to the direction of the flow. The fluid is subjected to a localized suction and is considered as a radiative optically thin gray fluid. The Reynolds Averaged Boundary Layer (RABL) equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The nonlinear and coupled system of partial differential equations (PDEs) is solved using the Keller box method. For the eddy-kinematic viscosity the Baldwin Lomax turbulent model and for the turbulent Prandtl number the extended Kays Crawford model are used. The numerical results show that the flow field can be controlled by the combined effect of the applied magnetic field, thermal radiation, and localized suction, moving the separation point, xs , downstream towards the plate’s end, and increasing total drag, D . The combined effect of thermal radiation and magnetic field has a cooling effect on the fluid at the wall vicinity. The combined effect has a greater influence in the case of high free-stream temperature.

Radiation effect on the turbulent compressible boundary layer flow with adverse pressure gradient

The effects of thermal radiation and localized suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient are numerically studied. The compressible flow is subjected to a constant localized suction velocity and the fluid is considered as a radiative optically thin gray fluid. The plate is adiabatic and the flow is subjected to adverse pressure gradient.The Reynolds Averaged Boundary Layer (RABL) equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The resulting nonlinear, coupled system of partial differential equations (PDEs) is solved using the Keller box method. For the eddy kinematic viscosity, the turbulent models of Cebeci Smith and Baldwin Lomax are employed. For the turbulent Prandtl number, the extended Kays Crawford model is used. The obtained results are validated with previously published computational results and with experimentally based correlations, showing a good agreement.The results show that the flow is influenced by the combined effect of radiation and localized suction. The coupled effect moves the separation point downstream, towards the plate’s end, and increases total drag. Radiation alters the thermal boundary layer above the plate by cooling the fluid at plate’s vicinity and by slightly increasing the boundary layer maximum temperature.

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.

Similarity solutions of compressible flow over a rotating cone with surface suction

We present solutions of the laminar compressible boundary-layer flows over the family of rotating cones subject to surface mass flux. The work is a generalization of previous studies of the compressible rotating-disk flow and incompressible rotating-cone flow without surface mass flux. Transformations are used which lead to a system of generalized von Karman equations with boundary conditions parameterized by half-angle and a mass-flux parameter. Results are discussed in terms of wall temperature and local Mach number in the particular case of air, although the formulation is readily extended to other fluids. It is suggested that suction acts a stabilizing mechanism, whereas increased wall temperature and local Mach number have destabilizing influences.

A new inviscid mode of instability in compressible boundary-layer flows

The stability of an almost inviscid compressible fluid flowing over a rigid heated surface is considered. We focus on the boundary layer that arises. The effect of surface heating is known to induce a streamwise acceleration in the boundary layer near the surface. This manifests in a streamwise velocity which exhibits a maximum larger than the free-stream velocity (i.e. the streamwise velocity exhibits an ‘overshoot’ region). We explore the impact of this overshoot on the stability of the boundary layer, demonstrating that the compressible form of the classical Rayleigh equation (which governs the development of short wavelength instabilities) possesses a new unstable mode that is a direct consequence of this overshoot. The structure of this new class of modes in the small wavenumber limit is detailed, providing a valuable confirmation of our numerical results obtained from the full inviscid eigenvalue problem.