The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

Shock wave/boundary layer interactions occur in a wide range of supersonic internal and external flows, and often these interactions are associated with turbulent boundary layer separation. The resulting separated flow is associated with large-scale, low-frequency unsteadiness whose cause has been the subject of much attention and debate. In particular, some researchers have concluded that the source of low-frequency motions is in the upstream boundary layer, whereas others have argued for a downstream instability as the driving mechanism. Owing to substantial recent activity, we are close to developing a comprehensive understanding, albeit only in simplified flow configurations. A plausible model is that the interaction responds as a dynamical system that is forced by external disturbances. The low-frequency dynamics seem to be adequately described by a recently proposed shear layer entrainment-recharge mechanism. Upstream boundary layer fluctuations seem to be an important source of disturbances, but the evidence suggests that their impact is reduced with increasing size of the separated flow.

Low-Frequency Unsteadiness of Shock Wave/Turbulent Boundary Layer Interactions

https://www.colonius.caltech.edu/pdfs/ColoniusLele2004.pdf

Computational aeroacoustics: progress on nonlinear problems of sound generation

A spatially developing supersonic adiabatic flat plate boundary layer flow (at M∞=2.25 and Reθ≈4000) is analyzed by means of direct numerical simulation. The numerical algorithm is based on a mixed weighted essentially nonoscillatory compact-difference method for the three-dimensional Navier–Stokes equations. The main objectives are to assess the validity of Morkovin’s hypothesis and Reynolds analogies, and to analyze the controlling mechanisms for turbulence production, dissipation, and transport. The results show that the essential dynamics of the investigated turbulent supersonic boundary layer flow closely resembles the incompressible pattern. The Van Driest transformed mean velocity obeys the incompressible law-of-the-wall, and the mean static temperature field exhibits a quadratic dependency upon the mean velocity, as predicted by the Crocco–Busemann relation. The total temperature has been found not to be precisely uniform, and total temperature fluctuations are found to be non-negligible. Consiste...

Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M=2.25

The interaction of a spatially developing adiabatic boundary layer flow at M∞=2.25 and Reθ=3725 with an impinging oblique shock wave (β=33.2°) is analyzed by means of direct numerical simulation of the compressible Navier-Stokes equations. Under the selected flow conditions the incoming boundary layer undergoes mild separation due to the adverse pressure gradient. Coherent structures are shed near the average separation point and the flow field exhibits large-scale low-frequency unsteadiness. The formation of the mixing layer is primarily responsible for the amplification of turbulence, which relaxes to an equilibrium state past the interaction. Complete equilibrium is attained in the inner part of the boundary layer, while in the outer region the relaxation process is incomplete. Far from the interaction zone, turbulence exhibits a universal behavior and it shows similarities with the incompressible case. The interaction of the coherent structures with the incident shock produces acoustic waves that prop...

Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20 ◦ with respect to the wall, and with an overall size of approximately 30 dissipative length scales.

Characterization of coherent vortical structures in a supersonic turbulent boundary layer

The interaction of a normal shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number M∞ = 1.3 and Reynolds number Reθ ≈ 1200 (based on the momentum thickness of the upstream boundary layer) is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations. The computational methodology is based on a hybrid linear/weighted essentially non-oscillatory conservative finite-difference approach, whereby the switch is controlled by the local regularity of the solution, so as to minimize numerical dissipation. As found in experiments, the mean flow pattern consists of an upstream fan of compression waves associated with the thickening of the boundary layer, and the supersonic region is terminated by a nearly normal shock, with substantial bending of the interacting shock. At the selected conditions the flow does not exhibit separation in the mean. However, the interaction region is characterized by ‘intermittent transitory detachment’ with scattered spots of instantaneous flow reversal throughout the interaction zone, and by the formation of a turbulent mixing layer, with associated unsteady release of vortical structures. As found in supersonic impinging shock interactions, we observe a different amplification of the longitudinal Reynolds stress component with respect to the others. Indeed, the effect of the adverse pressure gradient is to reduce the mean shear, with subsequent suppression of the near-wall streaks, and isotropization of turbulence. The recovery of the boundary layer past the interaction zone follows a quasi-equilibrium process, characterized by a self-similar distribution of the mean flow properties.

Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation

In the present paper the statistical properties of compressible isotropic turbulence are analyzed by means of direct numerical simulations. The scope of the work is to evaluate the influence of compressibility on the time evolution of mean turbulence properties and to quantify the statistical properties of turbulent structures, their dynamics and similarities with the incompressible case. Simulations have been carried out at various turbulent Mach numbers and compressibility ratios by using a conservative hybrid scheme that relies on an optimized weighted essentially nonoscillatory approach for the convective terms and compact differencing for the viscous contributions. In order to identify similarities with incompressible turbulence we have also carried out an analysis in the plane of the second (Q*) and third (R*) invariants of the anisotropic part of the deformation rate tensor. The simulations show that the joint probability density function (Q*,R*) has a universal structure, as found in incompressibl...

Francesco Grasso

Papers

Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M=2.25

Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M=2.25

Characterization of coherent vortical structures in a supersonic turbulent boundary layer

Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation

Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures