A two-dimensional body that moves suddenly in a viscous fluid can instantly generate vortices at its sharp edges. Recent work using inviscid flow theory, based on the Birkhoff–Rott equation and the Kutta condition, predicts that the ‘starting vortices’ generated by the sharp and straight edges of a body – i.e. the vortices formed immediately after motion begins – can be one of three distinct self-similar types. We explore the existence of these starting vortices for a flat plate and two symmetric Joukowski aerofoils immersed in a viscous fluid, using high-fidelity direct numerical simulations (DNS) of the Navier–Stokes equations. A lattice Green's function method is employed and simulations are performed for chord Reynolds numbers ranging from 5040 to 45 255. Vortices generated at the leading and trailing edges of the flat plate show agreement with the derived inviscid theory, for which a detailed assessment is reported. Agreement is also observed for the two symmetric Joukowski aerofoils, demonstrating the utility of the inviscid theory for arbitrary bodies. While this inviscid theory predicts an abrupt transition between the starting-vortex types, DNS shows a smooth transition. This behaviour occurs for all Reynolds numbers and is related to finite-time effects – there is a maximal time for which the (self-similar) starting vortices exist. We confirm the inviscid prediction that the leading-edge starting vortex of a flat plate can be suppressed dynamically. This has implications for the performance of low-speed aircraft such as model aeroplanes, micro air vehicles and unmanned air vehicles.

We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette-Poiseuille flow. A particular example is specified by spherical coordinates (Re, theta, phi), where Re = 6000 is a global Reynolds number, f denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and tan. specifies the ratio of the mean volume-flow speed to the plate speed. The limits phi -> 0 degrees and phi -> 90 degrees give alignment and orthogonality, respectively, while theta -> 0 degrees, theta -> 90 degrees correspond respectively to pure Couette flow in the x direction and pure Poiseuille flow at angle phi to the x axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of (Re, theta, phi). The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with theta = 90 degrees. With increasing theta, the energy peak gradually moves in the direction of increasing k(x) and decreasing k(z). Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the Q-criterion visualization of a three-dimensional time-averaged flow.

The force and torque on a solid body in a viscous potential flow are often taken to be independent of viscosity. Joseph et al. (Eur. J. Mech. B/Fluids, vol. 12, 1993, pp. 97–106; J. Fluid Mech., vol. 265, 1994, pp. 1–23) proved that this holds for (i) the force (not the torque) in any two-dimensional flow, and (ii) the drag force experienced by a purely translating three-dimensional body. The remaining components of the force and torque, along with general three-dimensional flows, were not considered. Importantly, the flow was assumed to be unbounded and irrotational everywhere. We eliminate this rarely satisfied assumption and consider the viscous force and torque experienced by any closed surface where the flow is irrotational locally; this can include a body's surface. Any vorticity distribution is permitted away from the closed irrotational surface. In so doing, we complete the analysis of Joseph et al. for all components of the viscous force and torque in two and three dimensions and enable application to real flows that inevitably contain regions of vorticity.

The starting vortex generated at the trailing edge of a flat plate, that is impulsively translated at fixed angle of attack, is a widely studied canonical problem. Recent work that examined the effect of plate rotation on this starting vortex found that two new and distinct vortex sheet types can arise. We generalise this work to study the starting vortex generated at any sharp and straight edge of an arbitrary body under a general time-dependent two-dimensional motion. The dimensionless velocity field of the attached flow near any sharp edge is assumed to take the form, $\hat {z}^{-1/2} f(T) + g(T) + o (1)$ , where $\hat {z}$ is the dimensionless position referenced to the edge, $f(T)$ and $g(T)$ are functions of dimensionless time, $T$ , associated with the local flow perpendicular and parallel to the edge, respectively. This enables starting vortices to be generally calculated and their types related by simply inspecting the forms of $f(T)$ and $g(T)$ . We elucidate the physics underlying all three vortex types and show that these vortices are generated by pure translation of the sharp edge. Several case studies are explored, including the leading/trailing edge vortices of a flat plate which can simultaneously be of different type (relevant to low-speed aircraft), the vortex formed by translation of a semi-infinite flat plate and the trailing-edge vortex of Joukowski aerofoils. With the ability to calculate the vortices at all edges, the theory is used to develop a general formula for the lift force of a flat plate which can find application in practice.

We discuss the modal, linear stability analysis of generalized Couette-Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle 0 <= phi <= 90 degrees to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number Re, phi and an angle 0 <= phi <= 90 degrees, where cos theta is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers (alpha, beta). With (theta, phi) fixed, the neutral surface f (theta, phi; Re, alpha, beta) = 0 in (Re, alpha, beta) space is discussed. A mapping from GCP to plane Couette-Poiseuille flow stability is found that suggests a scaling relation Re* alpha/k = H(theta*) that collapses all critical parameters, where Re* = Re (alpha(1)/alpha) (sin theta/sin theta*) and tan theta* = (alpha(1)/alpha) tan., with alpha(1) = alpha cos phi + beta sin phi. This analysis does not, however, directly reveal global critical properties for GCP flow. The global Re-cr(theta, phi) shows continuous variation, while alpha(cr)(theta, f) and beta(cr)(theta, phi) show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in Re) or lobes of the neutral surface. The discontinuity behaviour exists for all low phi. For phi greater than or similar to 21 degrees, variations of alpha(cr)(theta) and beta(cr)(theta) are generally smooth and monotonic.

We consider the evolution of a spatially periodic, perturbed vortex sheet for small times after the formation of a curvature singularity at time $t=t_c$ as demonstrated by Moore ( Proc. R. Soc. Lond. A, vol. 365, issue 1720, 1979, pp. 105–119). The Moore analysis is extended to provide the small-amplitude, full-sheet structure at $t=t_c$ for a general single-mode initial condition in terms of polylogarithmic functions, from which its asymptotic form near the singular point is determined. This defines an intermediate evolution problem for which the leading-order, and most singular, approximation is solved as a Taylor-series expansion in $\tau = t-t_c$ , where coefficients are calculated by repeated differentiation of the defining Birkhoff–Rott (BR) equation. The first few terms are in good agreement with numerical calculation based on the full-sheet solution. The series is summed, providing an analytic continuation which shows sheet rupture at circulation $\varGamma =0^+$ , $\tau >0^+$ , but with non-physical features owing to the absence of end-tip sheet roll up. This is corrected by constructing an inner solution with $\varGamma < \tau$ , as a perturbed similarity form with small parameter $\tau ^{1/2}$ . Numerical solutions of both the inner, nonlinear zeroth-order and first-order linear BR equations are obtained whose outer limits match the intermediate solution. The composite solution shows sheet tearing at $\tau =0^+$ into two separate, rolled up algebraic spirals near the central singular point. Branch separation distance scales as $\tau$ with a non-local, $\tau ^{3/2}$ correction. Properties of the intermediate and inner solutions are discussed.

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$ , where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$ , wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$ , and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$ . Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$ . The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$ , $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$ , $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$ , while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$ .

A new approach based on the power mean is suggested for defining an equivalent homogeneous roughness length kehr which takes into account patchiness or heterogeneous distribution of roughness on ship hulls and can be readily incorporated into existing full-scale drag prediction methods. In the limit where patch sizes are much greater than the boundary layer thickness, it is readily shown that the relationship between drag coefficient and roughness length is non-linear, highlighting an obvious source of error with current approaches that attempt to define an equivalent homogeneous roughness through an area-weighed arithmetic mean. The degree of error is dependent on the roughness distribution, but is estimated to exceed 16% for highly skewed beta heterogeneous distributions. For fully-rough models, the power-mean approach returns errors of <1% for the distributions tested here. The efficacy of the power-mean approach is also evaluated in the transitional regime and with different transitional roughness models (Nikuradse and Colebrook) and retains accuracy for most realistic operating scenarios.

We describe wall-resolved and wall-modelled large-eddy simulation (LES) of plane Couette (PC) flow. Subgrid-scale (SGS) motion is represented using the stretched-spiral vortex SGS model and the virtual wall model is employed for wall-modelled LES. Cases studied include direct numerical simulation (DNS) at friction Reynolds numbers $Re_\tau = 220$ , wall-resolved LES at $Re_\tau \sim 500\text {--}3600$ and wall-modelled LES at $Re_\tau \sim 3600\text {--}2.8\times 10^{5}$ . All LES performed show the presence of approximately spanwise periodic sets of streamwise rolls. Averaged (including spanwise) wall-normal profiles of the mean streamwise velocity show a consistent log region across all Reynolds numbers. Two distinct measures of turbulent intensity are explored, one of which recognizes the roll structure and one that does not. The spanwise variation of turbulence flow metrics is investigated. Mean streamwise velocity profiles show substantial spanwise variation but collapse well when normalized by local skin-friction velocities. Similar collapse is found for streamwise turbulent intensities. For all present LES, the mean skin-friction variation with the plate Reynolds number is found to match a simple analytical form (Pirozzoli et al., J. Fluid Mech., vol. 758, 2014, pp. 327–343) while the scaled centre-plane, mean-velocity gradient exhibits an inverse ProductLog dependence. Both the mean-flow roll energy and circulation, scaled with outer variables, decrease monotonically for $Re_\tau \gtrsim 500$ . At lower $Re_\tau$ , the mean streamwise zero-velocity line follows a wavy form in the spanwise direction, while at our larger $Re_\tau$ , a mushroom shape emerges which could potentially enhance local momentum transport in the spanwise direction and be responsible for the weakening of the spanwise rolls.

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thin-layer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The three-curve equations of contour dynamics that fully describe the double-layer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initial-value evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the single-layer equation of Moore ( Stud. Appl. Math. , vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the first-order equations to small-amplitude perturbations shows Kelvin–Helmholtz instability when the far-field fluid velocities on either side of the double layer are unequal. Equal velocities define a circulation-free double vorticity layer, for which solution of the initial-value problem using the Laplace transform reveals a double pole in transform space leading to linear algebraic growth in general, but there is a class of interesting initial conditions with no linear growth. This is shown to agree with the long-wavelength limit of the full linearized, three-curve stability equations.

This study presents an experimental dataset documenting the evolution of a turbulent boundary layer downstream of a rough-to-smooth surface transition. To investigate the effect of upstream flow conditions, two groups of experiments are conducted. For the Group-Re cases, a nominally constant viscous-scaled equivalent sand grain roughness k(s0)(+) approximate to 160 is maintained on the rough surface, while the friction Reynolds number Re-tau 0 ranges from 7100 to 21 000. For the Group-ks cases, Re-tau 0 approximate to 14 000 is maintained while k(s0)(+) ranges from 111 to 228. The wall-shear stress on the downstream smooth surface is measured directly using oil-film interferometry to redress previously reported uncertainties in the skin-friction coefficient recovery trends. In the early development following the roughness transition, the flow in the internal layer is not in equilibrium with the wall-shear stress. This conflicts with the common practise of modelling the mean velocity profile as two log laws below and above the internal layer height, as first proposed by Elliott (Trans. Am. Geophys. Union, vol. 39, 1958, pp. 1048-1054). As a solution to this, the current data are used to model the recovering mean velocity semi-empirically by blending the corresponding rough-wall and smooth-wall profiles. The over-energised large-scale motions leave a strong footprint in the near-wall region of the energy spectrum, the frequency and magnitude of which exhibit dependence on Re-tau 0 and k(s0)(+), respectively. The energy distribution in near-wall small scales is mostly unaffected by the presence of the outer flow with rough-wall characteristics, which can be used as a surrogate measure to extract the local friction velocity.

We consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude . We develop a complex-variable formulation for a nonlinear theory of generalized geometrical shock dynamics (GGSD) (Best, Shock Waves, vol. 1, issue 4, 1991, pp. 251-273; Best, Proc. R. Soc. Lond. A, vol. 442, 1993, pp. 585-598) as a hierarchical expansion of the Euler equations that can be closed at any order. The zeroth-order truncation of GGSD is related to the equations of Whitham's geometrical shock dynamics (GSD), while higher-order corrections incorporate non-uniformity of the flow immediately behind the piston-driven shock. Numerical solutions to GGSD systems up to second order are coupled to an edge-detection algorithm in order to investigate the hypothesized development of a shock-shape curvature singularity as the rippled shock evolves. This singular behaviour, together with the simultaneous development of a Mach-number discontinuity, is found at all orders of the GGSD hierarchy for both weak and strong shocks. The critical time at which a curvature singularity occurs converges as the order of the GGSD system increases at fixed , and follows a scaling inversely proportional to at sufficiently small values. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. (J. Fluid Mech., vol. 846, 2018, pp. 536-562) for a general Mach-number perturbation on a planar shock, and suggests that this represents the universal behaviour of a slightly perturbed, planar shock.

We perform direct numerical simulation and large-eddy simulation of an initially spherical region of turbulence evolving in free space. The computations are performed with a lattice Green's function method, which allows the exact free-space boundary conditions to be imposed on a compact vortical region. Large-eddy simulations are conducted with the stretched vortex subgrid stress model. The initial condition is spherically windowed, isotropic homogeneous incompressible turbulence. We study the spectrum and statistics of the decaying turbulence and compare the results with decaying isotropic turbulence, including cases representing different low-wavenumber behaviour of the energy spectrum (i.e. k(2) versus k(4)). At late times the turbulent sphere expands with both mean radius and integral scale showing similar timewise growth exponents. The low-wavenumber behaviour has little effect on the inertial scales, and we find that decay rates follow the predictions of Saffman (J. Fluid Mech., vol. 27, 1967, pp. 581-593) in both cases, at least until approximately 400 initial eddy turnover times. The boundary of the spherical region develops intermittency and features ejections of vortex rings. These are shown to occur at the integral scale of the initial turbulence field and are hypothesized to occur due to a local imbalance of impulse on this scale.

The effect of an initially uniform magnetic field of arbitrary orientation on the Richtmyer–Meshkov instability in Hall-magnetohydrodynamics (MHD) and ideal MHD is considered. Attention is restricted to the case where the initial density interface has a single-mode sinusoidal perturbation in amplitude and is accelerated by a shock traveling perpendicular to the interface. An incompressible Hall-MHD model for this flow is developed by solving the relevant impulse-driven linearized initial value problem. The ideal MHD theory is naturally obtained by taking the limit of vanishing ion skin depth. It is shown that the out-of-plane magnetic field component normal to both the impulse and the interface perturbation does not affect the evolution of the flow. For all field orientations other than strictly out-of-plane, the growth of interface perturbations is suppressed. However, the suppression is most effective for near tangential fields but becomes less effective with increasing ion skin depth and Larmor radius. The modeled suppression mechanism is transport of vorticity along magnetic field lines via Alfvén fronts in ideal MHD, and via a dispersive wave system in Hall-MHD. Oscillation of the interface growth rate is caused by a continuous phase change of the induced velocities at the interface due to vorticity transport parallel to the perturbation direction in ideal MHD, while it can also result from interfacial vorticity production associated with the ion cyclotron effect in Hall-MHD with a finite Larmor radius. The limiting flow behavior of a large ion-skin-depth is explored. To assess the accuracy and appropriateness of the incompressible model, its ideal MHD predictions are compared to the results of the corresponding shock-driven nonlinear compressible simulations.

We present wall-resolved large-eddy simulations (LES) of the incompressible Navier-Stokes equations together with empirical modeling for {turbulent} Taylor-Couette {(TC)} flow where the inner cylinder is rotating with angular velocity $\Omega_i$ and the outer cylinder is stationary. A simple empirical model of the turbulent, TC flow is developed consisting of near-wall, log-like turbulent wall layers separated by an annulus of constant angular momentum. The model is closed by a proposed scaling relation concerning the thickness of the wall layer on the inner cylinder. Model results include the Nusselt number $Nu$ (torque required to maintain the flow) and various measures of the wall-layer thickness as a function of both the Taylor {number} $Ta$ and $\eta$. These agree reasonably with experimental measurements, direct numerical simulation (DNS) and the present LES over a range of both $Ta$ and $\eta$. In particular, the model shows that, at fixed $\eta<1$, $Nu$ grows like $Ta^{1/2}$ divided by the square of the Lambert, (or Product-Log) function of a variable proportional to $Ta^{1/4}$. This cannot be represented by a power law dependence on $Ta$. At the same time the wall-layer thicknesses reduce slowly in relation to the cylinder gap. This suggests an asymptotic, very large $Ta$ state consisting of constant angular momentum in the cylinder gap with $u_\theta = 0.5\,\Omega_i\,R_i^2/r$, where $r$ is the radius, with vanishingly thin turbulent wall layers at the cylinder surfaces. An extension of the model to rough-wall turbulent wall flow at the inner cylinder surface is described. This shows an asymptotic, fully rough-wall state where the torque is independent of $Re_i/Ta$, and where $Nu\sim Ta^{1/2}$.

We consider the trailing-edge vortex produced in an inviscid fluid by the start-up motion of a two-dimensional flat plate. A general starting motion is studied that includes the initial angle-of-attack of the plate (which may be zero), individual time power laws for plate translational and rotational speeds and the pivot position for plate rotation. A vortex-sheet representation for a start-up separated flow at the trailing edge is developed whose time-wise evolution is described by a Birkhoff-Rott equation coupled to an appropriate Kutta condition. This description includes convection by the outer flow, rotation and vortex-image self-induction. It admits a power-law similarity solution for the (small-time) primitive vortex, leading to an equation set where each term carries its own time-wise power-law factor. A set of four general plate motions is defined. Dominant-balance analysis of this set leads to discovery of three distinct start-up vortex-structure types that form the basis for all vortex motion. The properties of each type are developed in detail for some special cases. Numerical and analytical solutions are described and transition between solution types is discussed. Singular and degenerate vortex behaviour is discovered which may be due to the absence of fluid viscosity. An interesting case is start-up motion with zero initial angle of attack coupled to power-law plate rotation for which time-series examples are given that can be compared to high Reynolds number viscous flows.

We discuss large-eddy simulations (LES) of the incompressible Navier-Stokes equations for Taylor-Couette flow. The ratio of the two co-axial cylinder diameters is η = ri/ro = 0.909 with ri the inner cylinder radius and ro the outer radius. The outer cylinder is stationary while the inner cylinder rotates with constant angular velocity ωi. Subgrid stresses are represented using the stretched-vortex model (SVM) where the subgrid motion is modeled by subgrid vortices undergoing stretching by the local resolvedscale velocitygradient field. We report wall-resolved LES at Rei = driωi/ν up to Rei = 106 with ν the kinematic viscosity of the Newtonian fluid and d = ro− ri the cylinder gap. The present study focuses on the wall-turbulence behavior at relatively high Rei. Comparisons are made with direct numerical simulations (DNS) and with experimental results.

While planar shock waves are known to be stable to small perturbations in the sense that the perturbation amplitude decays over time, it has also been suggested that plane propagating shocks can develop singularities in some derivative of their geometry (Whitham (1974) Linear and nonlinear waves. Wiley, New York) in a nonlinear, wave reinforcement process. We present a spectral-based analysis of the equations of geometrical shock dynamics that predicts the time to singularity formation in the profile of an initially perturbed planar shock for general shock Mach number. We find that following an initially sinusoidal perturbation, the shock shape remains analytic only up to a finite, critical time that is a monotonically decreasing function of the initial perturbation amplitude. At the critical time, the shock profile ceases to be analytic, corresponding physically to the incipient formation of a “shock-shock” or triple point. We present results for gas-dynamic shocks and discuss the potential for extension to shock dynamics of fast MHD shocks.

We present explicit and implicit large eddy simulations for fully developed turbulent pipe flows using a continuous-Galerkin spectral element solver. On the one hand, the explicit stretched-vortex model (by Misra and Pullin [45] and Chung and Pullin [14]), accounts for an explicit treatment of unresolved stresses and is adapted to the high-order solver. On the other hand, an implicit approach based on a spectral vanishing viscosity technique is implemented. The latter implicit technique is modified to incorporate Chung & Pullin virtual-wall model instead of relying on implicit dissipative mechanisms near walls. This near-wall model is derived by averaging in the wall-normal direction and relying in local inner scaling to treat the time-dependence of the filtered wall-parallel velocity. The model requires space-time varying Dirichlet and Neumann boundary conditions for velocity and pressure respectively. We provide results and comparisons for the explicit and implicit subgrid treatments and show that both provide favourable results for pipe flows at Re-tau = 2 x 10(3) and Re-tau = 1.8 x 10(5) in terms of turbulence statistics. Additionally, we conclude that implicit simulations are enhanced when including the wall model and provide the correct statistics near walls. (C) 2019 Elsevier Ltd. All rights reserved.