Influence of Damping of a Compliant Surface on Inviscid Instability of Overlying Incompressible Boundary Layer

The instability of an incompressible boundary layer on a compliant plate with respect to inviscid perturbations is analyzed on the basis of triple-deck theory. It is shown that unstable inviscid perturbations persist only if the inertia of the plate is taken into account. It is found that an important role is played by the bending stiffness of the plate. Specifically, as it approaches a certain value, the instability can become arbitrarily high, but, with a further increase in the bending stiffness, it vanishes completely as soon as the bending stiffness reaches a threshold value.

This paper represents a continuation of the work by Tempelmannet al. (J. Fluid Mech., vol. 646, 2010b, pp. 5–37) on spatial optimal growth in incompressible boundary layers over swept flat plates. We present an extension of the methodology to compressible flow. Also, we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, despite stabilizing crossflow modes, destabilizes disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilizing effect on the boundary layer whereas concave walls have a destabilizing effect. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.

One of the major achievements in fluid mechanics in the last quarter of the twentieth century has been the development of an asymptotic description of perturbations to boundary layers known generally as 'triple deck theory'. These developments have had a major impact on our understanding of laminar fluid flow, particularly laminar separation. It is also true that the theory rests on three quarters of a century of development of boundary layer theory which involves analysis, experimentation and computation. All these parts go together, and to understand the triple deck it is necessary to understand which problems the triple deck resolves and which computational techniques have been applied. This book presents a unified account of the development of laminar boundary layer theory as a historical study together with a description of the application of the ideas of triple deck theory to flow past a plate, to separation from a cylinder and to flow in channels. The book is intended to provide a graduate level teaching resource as well as a mathematically oriented account for a general reader in applied mathematics, engineering, physics or scientific computation.

One of the major achievements in fluid mechanics in the last quarter of the twentieth century has been the development of an asymptotic description of perturbations to boundary layers known generally as 'triple deck theory'. These developments have had a major impact on our understanding of laminar fluid flow, particularly laminar separation. It is also true that the theory rests on three quarters of a century of development of boundary layer theory which involves analysis, experimentation and computation. All these parts go together, and to understand the triple deck it is necessary to understand which problems the triple deck resolves and which computational techniques have been applied. This book presents a unified account of the development of laminar boundary layer theory as a historical study together with a description of the application of the ideas of triple deck theory to flow past a plate, to separation from a cylinder and to flow in channels. The book is intended to provide a graduate level teaching resource as well as a mathematically oriented account for a general reader in applied mathematics, engineering, physics or scientific computation.

The stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing.

On the görtler instability of boundary layers

Spanwise wall oscillation is an easily implemented active control scheme. It can delay the transition to turbulent flows in incompressible boundary layers under appropriate control parameters. However, this delayed transition in compressible boundary layers has not yet been confirmed numerically. In this paper, we perform a series of direct numerical simulations of a spatially developing subsonic boundary layer with different spanwise wall oscillation periods and amplitudes. We numerically confirm the delay in the transition under appropriate control parameters. The delayed transition can be explained by the Stokes boundary layer induced by the spanwise wall oscillation affecting the streamwise vortices. The superposed spanwise velocity and streamwise vorticity observed in the Stokes boundary layer can weaken the development of instability in the original streamwise vortices under appropriate control parameters, leading to a delayed transition. Furthermore, the spanwise wall oscillation changes the flow compressibility, as can be inferred from the equation for the velocity divergence. The enhanced compressibility around the pre-transition regions also stabilizes the development of the streamwise vortices and causes a delayed transition.

The stability of an incompressible flat plate boundary layer with pressure gradient (Hartree pressure gradient parameter β = 1.0, 0.6, 0.4, 0.2, 0, −0.1, −0.14, and −0.1988) is computed from the linearized complete small disturbance equations. The analysis is nevertheless a quasiparallel treatment because although boundary layer growth is accounted for, the disturbance wave function is correct only in a strictly parallel flow. It is found that the nonparallel flow effect has a negligible influence on the critical Reynolds number Rδ*−c for 0.4 ≤ β ≤ 1.0, but leads to a decrease in Rδ*−c in the range −0.1988 ≤ β ≤ 0.4. The V terms lead to an increase in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to a decrease in the range −0.1988 ≤ β ≤ 0.4. The stream tube stretching term led to a decrease in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to an increase in the range 0.4 ≥ β ≥ −0.1988. The effect of the V terms and the stream tube stretching term ∂2U/∂x∂y appear to dominate all other boundary layer growth terms; near Rδ*−c for β = −0.1988 however, the stream tube stretching term and ∂2V/∂x2 appear to be of the same order of magnitude.

The mean parameters of a two-dimensional incompressible turbulent boundary layer with zero pressure gradient were measured on the smooth wall of the TsAGI T-128 transonic wind tunnel in the range Reθ = 5.3 × 104–2.9 × 105 velocity profiles, skin friction coefficient, and shape factor. Novel data were obtained in the range Reθ ≈ 2.35 × 105–2.9 × 105. At the maximum Reynolds number, the record value of the von Karman number Reτ ∼1 × 105 was reached and the logarithmic law was maintained up to y+ ≈ 1.3 × 104. The equilibrium state of the boundary layer was estimated using Clauser’s equilibrium parameter, which was G = 6.4–6.8 for high Reynolds numbers. The results of the investigation confirm the conclusions of other studies on the universality of dimensionless velocity profile in the boundary layer outer region. The results were compared with semi-empirical dependencies, direct numerical simulation (DNS) results and the results of other studies performed both on wind tunnel walls and on a flat plate. At high Reynolds numbers the agreement between the shape factor was within ΔH = ± 0.012 (± 1%), with a skin friction coefficient—Δcf= ± 0.00006 (± 3.7%).

Measurements have been performed of mean and fluctuating velocities as well as of Reynolds and wall shear stress in an incompressible axisymmetric turbulent boundary layer with a severe adverse pressure gradient. In the flow region where separation was incipient and, further downstream, where reverse flow occurred, only the x-components u and u′ and the skin friction were measured. The measurements were carried out by using conventional hot wires, pulsed wires and surface fences. A comparison of hot-wire and pulsed-wire measurements showed good agreement also at high local turbulence levels as long as the flow did not reverse; large discrepancies were found, however, close to those positions in the flow where the flow velocity changed sign and in the separated region of the boundary layer.

Direct numerical simulation of incompressible turbulent boundary layers and planar jets at high Reynolds numbers initialized with implicit large eddy simulation

Steady incompressible magnetohydrodynamic mixed convection boundary layer flow of a Casson fluid on an exponentially vertical shrinking sheet using the non‐Newtonian heating equation is investigated in this paper. There are three main objectives of this study, namely, to develop a new mathematical model, to obtain multiple solutions, and to perform stability analysis. The governing partial differential equations have been changed into nonlinear ordinary differential equations. The resultant equations of boundary value problems are then converted into the equivalent initial value problems using the shooting method before they can be solved using Runge‐Kutta of order four. The numerical results are obtained and found to be in good agreement with the published literature. The results also indicate that the velocity boundary layer becomes thinner as the magnetic, slip, and Casson parameters increase. Dual solutions for temperature and velocity distributions are obtained. Furthermore, the results suggest that the presence of the force of buoyancy (opposing flow case) would cause the occurrence of dual solutions. However, based on the stability analysis, only the first solution is stable.

The problem of finding the flow over a finite flat plate aligned with a uniform free stream is revisited. Multigrid is used to obtain accurate numerical solutions up to a Reynolds number of 4000. Fourier boundary conditions keep the computational domain small, with no loss of accuracy. Near the trailing edge, excellent agreement with first-order triple-deck theory is found. However, previous comparisons between computations, experiments, and triple-deck theory are shown to be misleading: In fact, triple-deck theory only accounts for half the drag excess (that part not due to the first-order Blasius boundary layer) even at R=4000. The remainder is shown to be due to, among other things, a large displacementlike effect in the boundary layer, i.e., an O(R−1) increase in skin friction extending over the whole plate.

Hot-wire anemometry measurements in an incompressible turbulent boundary-layer flow over a flat plate, at zero pressure gradient were made using two X-probes simultaneously. The over seven meters long flat plate installed within the wind tunnel generated an approximately 20-cm thick boundary layer, and R(theta) of about 4,000. The mean velocity and turbulence intensity data very close to the wall were measured with a single-wire. Space-time velocity fluctuation and shear-product correlation measurements enabled the calculation of large-scale structure convection velocity, inclination-angle to the wall, and spatial extent in space. The shear-product correlation results are reported for the first time.

An integral calculation method for incompressible three-dimensional boundary layers in axisymmetric diffusers