Wall-normal Velocity Research Articles

A driving mechanism of near-wall turbulence subject to adverse pressure gradient in a plane Couette flow

The effect of adverse pressure gradient (APG) on near-wall turbulence is studied, with a particular attention to the turbulence production mechanism. A plane turbulent Couette flow is considered for several values of constant APG in the lower wall region. A direct numerical simulation (DNS) in a large computational domain shows that turbulence near the lower wall continues to exist even at sufficiently large APGs. On increasing the APG, the cross-streamwise turbulence intensities increase, and the near-wall streaks gradually disappear. A linear analysis using the optimal transient growth indicates that the APG inhibits the generation of near-wall streaks due to the significant reduction of the mean shear in the region near the lower wall. The turbulent fluctuation dynamics beyond the linear regime is studied with a DNS in a minimal flow unit. The near-wall self-sustaining process involving streaks is significantly weakened or destroyed as APG increases, while the turbulent fluctuations become more isotropic and localised. Using a conditional averaging analysis, a new mechanism of near-wall turbulence production under strong APG is uncovered. This mechanism is initiated by the wall-normal nonlinear transport of an outer wall-normal velocity fluctuation to the near-wall region. The transported wall-normal velocity fluctuation is subsequently amplified via the Orr mechanism, resulting in the non-zero turbulence production involving spatially localised vortical structures. This mechanism is also confirmed by DNS of the flow in a large computational domain, where strong correlation between the wall-normal nonlinear transport and turbulence production is observed.

On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Proper Orthogonal Decomposition for a Turbulent Boundary Layer with Aero-Optic Distortion

The velocity field associated with the deflection of a laser beam passing through a heated incompressible boundary layer in the wall-normal direction was studied using variants of the classical proper orthogonal decomposition (POD) technique. Due to the heating of the boundary layer, temperature, density, and index of refraction fields were variable in the boundary layer and were instantaneously affected by turbulent coherent structures. The time-varying index of refraction field led to a time-varying beam path of a laser traversing the boundary layer; the final angle of the laser beam was measured simultaneously with particle image velocimetry. The relationship between coherent structures in the velocity field and the final beam angle was studied using conditional POD and space-time POD. Conditional POD was performed on the wall-normal velocity field using conditions of large upstream and downstream laser deflections. The modes from conditional POD were compared to modes from POD without conditions. Physical features within some conditional POD modes were found to be shifted in the streamwise direction compared to unconditioned modes, suggesting that the streamwise position of velocity features was correlated to laser deflection. Space-time POD resulted in the same modes as the conditional POD and showed the modes convecting in the streamwise direction without a significant change in the size of the structures. This suggested a hypothesis that the laser deflects when coherent structures reach particular positions relative to the laser position as the structures convect, aligning the largest local velocity and density gradients with the laser beam path.

Momentum flux fluctuations in wall turbulence: A formula beyond the law of the wall

Within wall turbulence, there is a sublayer where the mean wall-normal flux of the streamwise momentum is constant and related to the logarithmic wall-normal profile of the mean streamwise velocity. This relation, i.e., the law of the wall, has been used to estimate the mean stress at the wall surface. However, the momentum flux exhibits large temporal fluctuations. To relate them theoretically to those of the streamwise velocity at the same position from the wall, we consider an orthogonal decomposition of the fluctuations on a plane of the streamwise and wall-normal velocities. Since a large timescale is expected for the component that would dominate the momentum flux, it is singled out by temporal smoothing. The resultant formula is consistent with time-series data of a boundary layer in a wind tunnel. We also extend the formula to thermally stratified cases.

Near-wall Lamb vector and its temporal–spatial evolution in the viscous sublayer of wall-bounded flows

This paper gives the relation between the Lamb vector and fundamental surface quantities in the vicinity of a no-slip flat wall using the Taylor-series expansion solution of the Navier–Stokes equations for incompressible viscous flow. The wall-normal component of the Lamb vector is dominated by the boundary enstrophy at the first order and the boundary enstrophy flux at the second order. The tangential Lamb vector is contributed by the terms related to both the boundary vorticity divergence and the skin friction divergence. Then, the derived relation is validated in the three-dimensional nonorthogonal Hiemenz flow (an attachment line flow) and a single-phase turbulent channel flow simulated using the lattice Boltzmann method. For this Hiemenz flow where the boundary vorticity divergence vanishes, the skin friction divergence-related term dominates the distribution of the near-wall tangential Lamb vector. In the turbulent channel flow, both the skin friction divergence and boundary vorticity divergence-related terms have significant contributions to the streamwise component of the Lamb vector, which are associated with the strong wall-normal velocity events (SWNVEs) in the viscous sublayer. In contrast, the boundary vorticity divergence-related terms have the main contribution to the spanwise component of the Lamb vector. Furthermore, the temporal–spatial evolution of the kinetic energy of the Lamb vector (half of the inner product of the Lamb vector denoted by J) is studied. In the turbulent channel flow, the SWNVEs are the main contributors to the temporal–spatial evolution of J in the viscous sublayer. This evolution is dominated by the viscous dissipation effect due to the gradient of the Lamb vector and the coupling effect between a viscous source term and the Lamb vector. The relations presented in this paper could be useful in understanding the physical mechanisms of the initial formation and evolution of the Lamb vector in the viscous sublayer of wall-bounded turbulence.

Investigation of vortical structures in the wake of pseudo-random roughness surfaces in hypersonic reacting boundary-layer flows

The influence of roughness on the laminar-turbulent transition of hypersonic flows is investigated with Direct Numerical Simulations (DNS) in a re-entry scenario of an Apollo-like space capsule. A Mach 20 flow regime is simulated for conditions at an altitude of 57.5 km. This study uses two different generating shapes (sinusoidal and triangular shaped waves) for random distributed patches. Stronger streamwise vorticity is observed for most configurations with the triangular base functions. The simulation includes the chemical equilibrium gas model. Vortical structures in the wake of both roughness patch types interact with an acoustic perturbation derived from local one-dimensional inviscid (LODI) relations. A breakdown of the cross-flow-like vortex in the wake of the roughness patch is observed for both roughness patches. The growth of modal instabilities is studied with a spatio-temporal Fourier analysis. Both y- and z-type instabilities, which are present in cross-flow vortices at low speeds, are observed for the sinusoidal patch. In the triangular configuration, the y-type instability is the dominating instability which is in line with the stronger wall-normal velocity gradients caused by the increase in spanwise slope of the patch.

Physics-guided deep learning for generating turbulent inflow conditions

In this paper, we propose an efficient method for generating turbulent inflow conditions based on deep neural networks. We utilise the combination of a multiscale convolutional auto-encoder with a subpixel convolution layer ( ${\rm MSC}_{\rm {SP}}$ -AE) and a long short-term memory (LSTM) model. Physical constraints represented by the flow gradient, Reynolds stress tensor and spectral content of the flow are embedded in the loss function of the ${\rm MSC}_{\rm {SP}}$ -AE to enable the model to generate realistic turbulent inflow conditions with accurate statistics and spectra, as compared with the ground truth data. Direct numerical simulation (DNS) data of turbulent channel flow at two friction Reynolds numbers $Re_{\tau } = 180$ and 550 are used to assess the performance of the model obtained from the combination of the ${\rm MSC}_{\rm {SP}}$ -AE and the LSTM model. The model exhibits a commendable ability to predict instantaneous flow fields with detailed fluctuations and produces turbulence statistics and spectral content similar to those obtained from the DNS. The effects of changing various salient components in the model are thoroughly investigated. Furthermore, the impact of performing transfer learning (TL) using different amounts of training data on the training process and the model performance is examined by using the weights of the model trained on data of the flow at $Re_{\tau } = 180$ to initialise the weights for training the model with data of the flow at $Re_{\tau } = 550$ . The results show that by using only 25% of the full training data, the time that is required for successful training can be reduced by a factor of approximately 80% without affecting the performance of the model for the spanwise velocity, wall-normal velocity and pressure, and with an improvement of the model performance for the streamwise velocity. The results also indicate that using physics-guided deep-learning-based models can be efficient in terms of predicting the dynamics of turbulent flows with relatively low computational cost.

Linear instability and resonance effects in large-scale opposition flow control

Opposition flow control is a robust strategy that has been proved effective in turbulent wall-bounded flows. Its conventional set-up consists of measuring wall-normal velocity in the buffer layer and opposing it at the wall. This work explores the possibility of implementing this strategy with a detection plane in the logarithmic layer, where control could be feasible experimentally. We apply control on a channel flow at $Re_\tau = 932$ , only on the eddies with relatively large wavelengths ( $\lambda / h > 0.1$ ). Similarly to the buffer layer opposition control, our control strategy results in a virtual-wall effect for the wall-normal velocity, creating a minimum in its intensity. However, it also induces a large response in the streamwise velocity and Reynolds stresses near the wall, with a substantial drag increase. When the phase of the control lags with respect to the detection plane, spanwise-homogeneous rollers are observed near the channel wall. We show that they are a result of a linear instability. In contrast, when the control leads with respect to the detection plane, this instability is inactive and oblique waves are observed. Their wall-normal profiles can be predicted linearly as a response of the turbulent channel flow to a forcing with the advection velocity of the detection plane. The linearity, governing the flow, opens a possibility to affect large scales of the flow in a controlled manner, when enhanced turbulence intensity or mixing is desired.

High resolution simulation of film cooling with blowing ratio and inclination angle effects based on hybrid thermal lattice Boltzmann method

A series of high resolution simulations on film cooling with varying blowing ratios and inclination angles are carried out by using in-house code based on hybrid thermal lattice Boltzmann method. Three blowing ratios ranging from 0.2-0.8 and four inclination angles from 15? to 60? are chosen for the simulations. The evolutionary mechanism of coherent structure in three domains of film-covering region is studied from the view of space and time. Besides, the influencing mechanism of blowing ratio and inclination angle on flow and heat transfer features of film cooling is uncovered. Results show that hairpin vortex, hairpin packet, and quasi-streamwise vortex appearing in rotating domain play a key role in heat transfer process of film cooling. The strong ejection, sweep and vortex rotation resulted from these vortices enhance the convective heat transfer. It is also found that the size of coherent structure varies significantly with blowing ratio and its integral form shows a strong dependence on inclination angle. Moreover, inclination angle of coolant jet has a significant impact on turbulence fluctuation intensity. The influence of blowing ratio on the attachment of coolant film and film-cooling performance is more obvious than that of inclination angle. It is believed that all of these are related closely to the variation of streamwise and wall-normal jet velocity in the case of various blowing ratios and inclination angles.

波状壁の上を通過する粘弾性流体のせん断流れの数値解析

A two-dimensional shear flow of a viscoelastic fluid over a wavy wall was numerically analyzed by the finite difference method. The upper convected Maxwell (UCM) model was used as the constitutive equation, and the unsteady flow problem was solved numerically using the stream function-vorticity method. Variables are discretized using a staggered grid with uniform mesh width. The volume penalization method was used to represent the moving upper wall and the fixed lower wavy wall that induces a vorticity perturbation. The flow in an infinitely long channel is computed using periodic boundary conditions in the flow direction. The results show that the vorticity intensifies over the wavy surface with positive and negative vorticity stripes, at which extreme wall-normal velocity fluctuations are also observed. At relatively high Deborah numbers, the lower ends of these extreme fluctuation regions are close to the critical line associated with a change of type of the governing partial differential equations in a Couette flow. In contrast, the upper ends of the extreme fluctuation region are close to the critical layer of the shear flow, where the flow velocity coincides with the viscoelastic Alfvén wave speed.

Enhanced outer peaks in turbulent boundary layer using uniform blowing at moderate Reynolds number

Uniform blowing through a permeable surface acts as an active flow control method in wall bounded flows. Such control technique was investigated in a zero pressure gradient turbulent boundary layer over a flat plate. Measurement data were obtained with the help of Laser Doppler Anemometry technique. Besides the drag reduction characteristics of such flow control method, time averaged measurement of stream-wise and wall-normal velocity components was taken at Reynolds number based on momentum thickness () of 1100–3670. Due to the difference in surface condition with and without blowing, mean properties of the boundary condition at wall influence the flow properties when scaled with outer scaling properties. Enhanced turbulence is observed in Reynolds stresses using statistical analysis including the thickening of the boundary layer.

A method to determine wall shear stress from mean profiles in turbulent boundary layers

The direct measurement of wall shear stress in turbulent boundary layers (TBL) is challenging, therefore requiring it to be indirectly determined from mean profile measurements. Most popular methods assume the mean streamwise velocity to satisfy either a logarithmic law in the inner layer or a composite velocity profile with many tuned constants for the entire TBL, both of which require reliable data from the inner layer. The presence of roughness and pressure gradient brings additional complications where most existing methods either fail or require significant modification. A novel method is proposed to determine the wall shear stress in zero pressure gradient TBL from measured mean profiles, without requiring near-wall data. The method is based on the stress model of Kumar and Mahesh [Phys. Rev. Fluids 6, 024603 (2021)], who developed accurate models for mean stress and wall-normal velocity in zero pressure gradient TBL. The proposed method requires a single point measurement of mean streamwise velocity and mean shear stress in the outer layer, preferably between $20$ to $50 \%$ of the TBL, and an estimate of boundary layer thickness and shape factor. The method can handle wall roughness without modification and is shown to predict friction velocities to within $3 \%$ over a range of Reynolds number for both smooth and rough wall zero pressure gradient TBL. In order to include the pressure gradients effects, the work of Kumar and Mahesh [Phys. Rev. Fluids 6, 024603 (2021)] is revisited to derive a novel model for both mean stress and wall-normal velocity in pressure gradient TBL, which is then used to formulate a method to obtain the wall shear stress from the profile data. Overall, the proposed method is shown to be robust and accurate for a variety of pressure gradient TBL.

Experimental investigation of aerofoil tonal noise at low Mach number

This paper presents an experimental investigation of aerofoil tones emitted by a controlled-diffusion aerofoil at low Mach number ( $0.05$ ), moderate Reynolds number based on the chord length ( $1.4 \times 10^{5}$ ) and moderate incidence ( $5^{\circ }$ angle of attack). Wall-pressure measurements have been performed along the suction side of the aerofoil to reveal the acoustic source mechanisms. In particular, a feedback loop is found to extend from the aerofoil trailing edge to the regions near the leading edge where the flow encounters a mean favourable pressure gradient, and consists of acoustic disturbances travelling upstream. Simultaneous wall-pressure, velocity and far-field acoustic measurements have been performed to identify the boundary-layer instability responsible for tonal noise generation. Causality correlation between far-field acoustic pressure and wall-normal velocity fluctuations has been performed, which reveals the presence of a Kelvin–Helmholtz-type modal shape within the velocity disturbance field. Tomographic particle image velocimetry measurements have been performed to understand the three-dimensional aspects of this flow instability. These measurements confirm the presence of large two-dimensional rollers that undergo three-dimensional breakdown just upstream of the trailing edge. Finally, modal decomposition of the flow has been carried out using proper orthogonal decomposition, which demonstrates that the normal modes are responsible for aerofoil tonal noise. The higher normal modes are found to undergo regular modulations in the spanwise direction. Based on the observed modal shape, an explanation of aerofoil tonal noise amplitude reduction is given, which has been previously reported in modular or serrated trailing-edge aerofoils.

Features of surface physical quantities and temporal-spatial evolution of wall-normal enstrophy flux in wall-bounded flows

This paper presents a concise derivation of the temporal-spatial evolution equation of the wall-normal enstrophy flux on a no-slip flat wall. Each contribution to the evolution is explicitly expressed using the two fundamental surface quantities: skin friction (or equivalently surface vorticity) and surface pressure which are coupled through the boundary enstrophy flux (BEF). The newly derived relation is then used to explore, in a preliminary manner, the physical features of surface quantities and their dynamical roles in wall-bounded laminar and turbulent flows. It is confirmed that the BEF usually changes its sign near the separation and attachment lines in the skin friction field. For the simulated incompressible turbulent channel flow at Reτ=180, violent variations of different terms in the derived formulation are observed in the regions below the strong wall-normal velocity events (SWNVEs) when compared to other common regions. Near the SWNVEs, the evolution of the wall-normal enstrophy flux is found to be dominated by the wall-normal diffusion of the vortex stretching term which is relatively weak or negligible for laminar flows. Combined with our previous research results, it is conjectured that the strong interaction between the quasi-streamwise vortex and the channel wall intensifies the temporal-spatial evolution of the wall-normal enstrophy flux on the wall, which accounts for the highly intermittent feature of the viscous sublayer.

Turbulent flow in curved channels

Fully developed turbulent flow in channels with mild to strong longitudinal curvature is studied by direct numerical simulations. The Reynolds based on the bulk mean velocity and channel half-width $\delta$ is fixed at $20\,000$ , resulting in a friction Reynolds number of approximately 1000. Four cases are considered with curvature varying from $\gamma = 2\delta /r_c = 0.033$ to 0.333, where $r_c$ is the curvature radius at the channel centre. Substantial differences between the mean wall shear stress on the convex and concave walls are already observed for $\gamma = 0.033$ . A log-law region is absent and a region with nearly constant mean angular momentum develops in the channel centre for strong curvatures. Spanwise and wall-normal velocity fluctuations are strongly amplified by curvature in the outer region of the concave channel side. Only near the walls, where curvature effects are relatively weak, do the mean velocity and velocity fluctuation profiles approximately collapse when scaled by wall units based on the local friction velocity. Budgets of the streamwise and wall-normal Reynolds-stress equations are presented and turbulence structures are investigated through visualizations and spectra. In the case with strongest curvature, the flow relaminarizes locally near the convex wall. On the concave channel side, large elongated streamwise vortices reminiscent of Taylor–Görtler vortices develop for all curvatures considered. The maximum in the premultiplied two-dimensional wall-normal energy spectrum and co-spectrum shifts towards larger scales with increasing curvature. The large scales substantially contribute to the wall-normal velocity fluctuations and momentum transport on the concave channel side.

The ultimate state of turbulent permeable-channel flow

Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow with constant bulk mean velocity and temperature, $u_{b}$ and $\theta _{b}$ , between parallel, isothermal, no-slip and permeable walls. The wall-normal transpiration velocity on the walls $y=\pm h$ is assumed to be proportional to the local pressure fluctuations, i.e. $v=\pm \beta p/\rho$ (Jiménez et al., J. Fluid Mech., vol. 442, 2001, pp. 89–117). The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. Turbulent heat and momentum transfer in permeable-channel flow for the dimensionless permeability parameter $\beta u_b=0.5$ has been found to exhibit distinct states depending on the Reynolds number $Re_b=2h u_b/\nu$ . At $Re_{b}\lesssim 10^4$ , the classical Blasius law of the friction coefficient and its similarity to the Stanton number, $St\approx c_{f}\sim Re_{b}^{-1/4}$ , are observed, whereas at $Re_{b}\gtrsim 10^4$ , the so-called ultimate scaling, $St\sim Re_b^0$ and $c_{f}\sim Re_b^0$ , is found. The ultimate state is attributed to the appearance of large-scale intense spanwise rolls with the length scale of $O(h)$ arising from the Kelvin–Helmholtz type of shear-layer instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_b)$ as in free shear layers, so that the Taylor dissipation law $\epsilon \sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_b^0$ ) holds. In spite of strong turbulence promotion there is no flow separation, and thus large-amplitude temperature fluctuations of $O(\theta _b)$ can also be induced similarly. As a consequence, the ultimate heat transfer is achieved, i.e. a wall heat flux scales with $u_{b}\theta _{b}$ (or equivalently $St\sim Re_b^0$ ) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.

Mechanisms of instability growth, interaction and breakdown induced by a backward-facing step in a swept-wing flow

Time-resolved particle image velocimetry measurements were performed downstream of a swept backward-facing step. The measurements allow detailed analysis of the interactions between the unsteady instabilities and the stationary crossflow vortices. Different mechanisms are identified that lead to the modulation of the different families of unsteady instabilities that occur downstream of the step. For the low-frequency spanwise-travelling mode, the modulation occurs due to a redistribution of momentum when the instability encounters regions of large spanwise shear of the wall-normal and streamwise velocity. However, the higher-frequency streamwise-travelling instabilities undergo the familiar ‘lift-up’ mechanism when they encounter the regions of large vertical velocity due to the presence of the stationary crossflow vortices. The process leading to large velocity spikes, and ultimately to a laminar breakdown to turbulence, is identified as a constructive interaction between the different unsteady instabilities, coupled with an interaction with the stationary crossflow vortices when the phases align properly.

Layered structure of turbulent plane wall jet

This paper investigates the layered structure of a turbulent plane wall jet at a distance from the nozzle exit. Based on the force balances in the mean momentum equation, the turbulent plane wall jet is divided into three regions: a boundary layer-like region (BLR) adjacent to the wall, a half free jet-like region (HJR) away from the wall, and a plug flow-like region (PFR) in between. In the PFR, the mean streamwise velocity is essentially the maximum velocity, and the simplified mean continuity and mean momentum equations result in a linear variation of the mean wall-normal velocity and Reynolds shear stress. In the HJR, as in a turbulent free jet, a proper scale for the mean wall-normal flow is the mean wall-normal velocity far from the wall and a proper scale for the Reynolds shear stress is the product of the maximum mean streamwise velocity and the velocity scale for the mean wall-normal flow. The BLR region can be divided into four sub-layers, similar to those in a canonical pressure-driven turbulent channel flow or shear-driven turbulent boundary layer flow. Building on the log-law for the mean streamwise velocity in the BLR, a new skin friction law is proposed for a turbulent wall jet. The new prediction agrees well with the correlation of Bradshaw and Gee (1960) over moderate Reynolds numbers, but gives larger skin frictions at higher Reynolds numbers.

The Wake Structure of a Stationary and a Rotary Wall-Mounted Low-Aspect-Ratio Cylinder

This paper investigates the wake structure of a low-aspect-ratio rotating cylinder mounted wall-normal to a flat plate at a diameter-based Reynolds number of . The cylinder is driven at a constant rotational rate about its respective axial coordinate. Four nondimensional rotation rates were selected: 0.5, 1, 2, and 3, representing rates based on the static cylinder’s natural shedding frequency and corresponding harmonics. Two aspect ratios were investigated: i) and ii) . The case of represents a cylinder height equal to the local boundary-layer thickness. Both streamwise and cross-stream (transverse) particle image velocimetry (PIV) planes are presented, and the resulting time-averaged flowfield is examined. Results for the rotating cylinder reveal the development of a dominant streamwise vortex that persists for several length scales downstream of the cylinder. Accompanying the streamwise vortex formation is an enhancement in the wall-normal velocity component (downwash) behind the rotating cylinder. Additionally, the maximum downwash occurs immediately behind the rotating cylinder near 0.27 diameters downstream in contradistinction to a stationary cylinder with peak downwash near 2–3 diameters. The maximum downwash, being in line with the formation of the dominant streamwise vortex, shortens the recirculation region in the cylinder’s wake, thus shifting the plate reattachment point further upstream. In the case of , where the cylinder is submerged within the boundary layer, although the length of the recirculation region reflected by the mean flow is lengthened by the rotary motion, the maximum downwash for the rotating cylinder is enhanced by a factor of 10 from that of the maximum downwash experienced by a static counterpart.

Inviscid instability of compressible exponential boundary layer flows

Based on the linearized Euler equations and the normal-mode approach, the Pridmore-Brown equation for acoustics and the stability of parallel shear flows is solved for a boundary layer flow with an exponential velocity profile. The general solution is derived in terms of the confluent Heun function, and in turn, using the boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity, the boundary value problem is converted to an algebraic eigenvalue problem. Solutions to the eigenvalue problem allow a comprehensive picture of the stability of compressible boundary layers. In particular, the complex eigenvalues ω are calculated as a function of the streamwise wavenumber α and the Mach number M. We observe that with growing wavenumbers, the number of eigenvalues increases discretely. Only for M > 1, unstable modes exist, and the boundary between neutrally stable and unstable modes is defined by the transonic line ω = α to be calculated from a degenerated eigenvalue equation. For large wall distances, the eigenfunctions converge exponentially toward zero. The spatial decay rate defines an “acoustic boundary layer thickness” δa, which indicates how far outside modes are still audible. For M > 1 and large α, δa diverges exponentially, i.e., in this parameter range, modes are perceptible even far from the boundary layer. A particularly steep rise of δa is observed when M > 2. Thus, there is a wavenumber range in which ωi and δa are both large and thus generates a particularly strong noise impact. From the eigenfunction for the unstable modes, a strong increase in the amplitude of acoustic waves can be identified in the vicinity of the wall, which indicates an accumulation and saturation of energy and thereafter leads to temporal instability and sound radiation in the free stream.