Self-similar Boundary Layer Research Articles

Publisher's Note: “Scaling of self-similar turbulent boundary layers under adverse pressure gradient” [Phys. Fluids 36, 085204 (2024)]

Publisher's Note: “Scaling of self-similar turbulent boundary layers under adverse pressure gradient” [Phys. Fluids <b>36</b>, 085204 (2024)]

Scaling of self-similar turbulent boundary layers under adverse pressure gradient

We present a theoretical and numerical study of the self-similarity turbulent boundary layers (SSTBLs) under adverse pressure gradient. Two relations are emphasized in theoretical extension. The first one explains the necessary relation to establish the same TBL with different choices of scaling, called coordinate transformation relation. The second relation is the equivalent requirement on the outer scaling of the mean velocity profile, named as scaling equivalence relation. Numerical verification is implemented with a set of ideal SSTBL velocity profiles that exhibit no Reynolds number effect by numerically solving the generalized governing equations of SSTBL under specified external conditions and self-similar scalings. Existing scalings in literature and two presently designed scalings are examined based on self-similar requirement to show their fulfillment of the two relations. Large-eddy simulations (LESs) of SSTBL are performed by directly specifying the pressure gradient parameter β corresponding to different scalings. We show once the coordinate transformation relation is satisfied, the similar fully developed SSTBL can be settled with different scalings. Further detailed quantitative analysis confirms the validity of the two conditions. Two velocity profile models of SSTBL are evaluated with LES results. The former is based on the numerical solution of the SSTBL equations, and the latter is based on the similarity between the outer region of TBL and the laminar boundary layer. Both models work reasonably well for present LES.

The Planar Mixing Layer Flame (PMLF): a canonical configuration for studying non-premixed combustion chemistry and soot inception

The study of fuel chemistry and soot inception in non-premixed combustion can be advanced by characterizing flame configurations in which the advection and diffusion transport can be finely controlled, with the ability to decouple pyrolysis from oxidation. Also, the ideal flames to be investigated should be perturbed minimally by probes and thick enough for sampling techniques to yield spatially resolved measurements of their structure. The Planar Mixing Layer Flame (PMLF) configuration introduced herein is established between a fuel and an oxidizer slot jet adjacent to each other and shielded from the ambient air by annularly co-flowing inert nitrogen. The PMLF flow is kept laminar and steady by an impinging flat plate equipped with a rectangular exhaust slit opening which anchors the position of the hot combustion products via buoyancy. The PMLF is accessible to sampling and its flow stability is preserved when using any tested probe. The experiments are complemented with 2D- Computational Fluid Dynamics (CFD) modeling with detailed chemical kinetics. The results demonstrate that the PMLF has a self-similar boundary layer structure whose horizontal cross-sections are equivalent to properly selected and equally thick 1D- Counterflow Flames (CFs). The equivalence allows for excellent predictions of the PMLF thermochemical structure characterized experimentally but at a small fraction of the 2D-CFD computational cost. The 1D-CF equivalence affects even aromatics less than twofold despite their kinetics being known to be very sensitive to the temperature field. Importantly, the PMLF thickness is several millimeters and grows at increasing HABs so that the equivalent 1D-CFs have strain rates as small as 7.0 /s which cannot be studied in CF experiments. As a result, the PMLF emerges as a promising canonical non-premixed flame configuration for studying flame chemistry and soot inception on time scales of tens of milliseconds typical of many combustion applications.

Linear stability analysis of hypersonic boundary layers computed by a kinetic approach: a semi-infinite flat plate at $$\varvec{4.5\le \mathrm{M}_\infty \le 9}$$

Linear stability analysis is performed using a combination of two-dimensional Direct Simulation Monte Carlo (DSMC) method for the computation of the basic state and solution of the pertinent eigenvalue problem, as applied to the canonical boundary layer on a semi-infinite flat plate. Three different gases are monitored, namely nitrogen, argon and air, the latter as a mixture of 79\% Nitrogen and 21\% Oxygen at a range of free-stream Mach numbers corresponding to flight at an altitude of 55km. A neural network has been utilised to predict and smooth the raw DSMC data; the steady laminar profiles obtained are in very good agreement with those computed by (self-similar) boundary layer theory, under isothermal or adiabatic wall conditions, subject to the appropriate slip corrections computed in the DSMC method. The leading eigenmode results pertaining to the unsmoothed DSMC profiles are compared against those of the classic boundary layer theory. Small quantitative, but no significant qualitative differences between the results of the two classes of steady base flows have been found at all parameters examined. The frequencies of the leading eigenmodes at all conditions examined are practically identical, while perturbations corresponding to the DSMC profiles are found to be systematically more damped than their counterparts arising in the boundary layer at the conditions examined, when the correct velocity slip and temperature jump boundary conditions are imposed in the base flow profiles; by contrast, when the classic no-slip boundary conditions are used, less damped/more unstable profiles are obtained, which would lead the flow to earlier transition. On the other hand, the DSMC profiles smoothed by the neural network are marginally more stable than their unsmoothed counterparts.

On exactly self-similar turbulent boundary layers and single-layer scaling

Turbulent boundary layers are traditionally thought to obey two layer scaling. However, it was shown by Dixit and Ramesh [J. Fluid Mech. 615, 445–475 (2008)] that the so-called sink flow turbulent boundary layer obeys a Blasius-like scaling all through the boundary layer. We make this single layer scaling as an operational definition for perfectly equilibrium flow and derive equations for a general class of equilibrium flows with permeable walls allowing suction and blowing of fluid. Using the above definition, constraints required to be satisfied by such equilibrium turbulent boundary layers are derived and discussed in detail. Using direct numerical simulations (DNS), we demonstrate the existence of such turbulent boundary layers, and various properties of these boundary layers are studied in detail. In particular, it is shown that the mean velocity profiles and other turbulent statistics display excellent similarity, and various boundary layer parameters in such boundary layers vary systematically with the magnitude of blowing/suction parameter. By extension, it is also demonstrated by DNS that higher order moments (up to third order moments) also display streamwise self-similarity. We believe that this is an important result where self-similarity is demonstrated for the first time for higher moments of fluctuating velocity. The non-universality of the log-law coefficients in favorable pressure gradient flows is already known [S. A. Dixit and O. N. Ramesh, J. Fluid Mech. 615, 445–475 (2008) and Chauhan et al., in Proceedings of 11th EUROMECH European Turbulence Conference on Turbulence, Porto, Portugal (Springer Proceedings in Physics, 2007), pp. 239–241]. We demonstrate their non-universal nature in equilibrium boundary layers with suction and blowing using the present direct numerical simulations. Remarkably, the ratio of Reynolds shear stress (u′v′¯) to streamwise (u′2¯) and spanwise turbulent intensity (w′2¯) in these boundary layers is constant over the log-region in the mean velocity profile even though u′2¯, w′2¯, and u′v′¯ by themselves show variation with y+ in the log region, indicative of the absence of inactive motion in these flows. It is shown that the dominance of active motion and more importantly the absence of inactive motion leads to a universality of scaling that manifests as a single layer Blasius scaling with self-similarity in such flows. Equivalently, it could also be seen as a single layer scaling with all variables all through the boundary layer. This would mean that the friction velocity scales the entire boundary layer, which could be taken to suggest equivalence of attached eddies with the active component of motion.

Dynamics of the jet wiping process via integral models

Abstract

Influence of surface sublimation on the stability of the supersonic boundary layer and the laminar–turbulent transition

We report a theoretical study of the properties of a supersonic boundary layer and its linear stability under conditions of surface material sublimation. Calculations were performed for an adiabatic boundary layer for a flat plate with a naphthalene coating at a free-stream Mach number of M = 3 (for the first instability mode disturbances). In the boundary layer, surface sublimation generates a binary mixture flow (air and foreign vapors). This flow is studied using local self-similar boundary layer equations, and it is shown that the rise in the flow stagnation temperature and the corresponding evaporation of the wall material cause significant wall cooling and an increase in the near-wall density of the binary mixture. This modification of the boundary layer profiles leads to a decrease in the disturbance amplification rates. This is confirmed by calculations based on linear stability theory (LST). Boundary-layer stabilization occurs with an increase in stagnation temperature. The influence of surface sublimation on the position of the laminar–turbulent transition was estimated by means of the LST-based eN method. The possibility of increasing the transition Reynolds number by application of the sublimation coating is demonstrated. The results of pilot boundary layer transition experiments performed in a hot-shot wind tunnel are reported. For the first time, a delay in the transition due to the application of a naphthalene coating was experimentally demonstrated. It is also shown that surface sublimation leads to an increase in the growth rates of the second and third instability modes for a Mach 8 boundary layer.

Reynolds analogy factor in self-similar compressible turbulent boundary layers with pressure gradients

Abstract

Unsteady separated stagnation-point flows and heat transfer over a plane surface moving normal to the flow impingement

In this paper, we have investigated the momentum and heat transfer in an unsteady separated stagnation-point flow of a viscous fluid over a plane surface advancing towards or receding from the normal stagnation flow which impinges on the surface with a variable strain rate s(t), causing the unsteadiness in this flow problem. The flow and heat transfer characteristics are therefore governed by the flow strength (stagnation) parameter a, unsteadiness parameter β, plate velocity (normal) parameter α and Prandtl number Pr. The current analysis ensures the existence of the self-similar solutions to this flow problem only when the plate velocity varies directly to the square root of the strain rate s(t). A closed form analytic solution of this flow problem is found for the specific relational values of β=2a. The solutions of this flow problem are non unique only under the negative values of β. When the plate surface is advancing towards the normal stagnation flow (i.e., when α> 0), the self-similar boundary layer solution continues for any given values of a and β, whereas for receding of the plate surface from the incoming flow (i.e., for α< 0) the boundary layer solution does not exist after a certain value of α depending upon the values of a and β. In the case where receding of the plate surface occurs, the features of the boundary layer flows depend highly on the sum values of the parameters a and β. For (a+β)> 0, the boundary layer solution ends with an attached flow solution, while it gets terminated at a reverse flow solution for (a+β)< 0. And for (a+β)= 0, the governing boundary layer equations provide us with the trivial (zero) solutions which are unable to capture the free boundary conditions.

Analysis of the spanwise extent and time persistence of uniform momentum zones in zero pressure gradient and adverse pressure gradient turbulent boundary layers

Time-resolved velocity fields from direct numerical simulations (DNS) are used to investigate the spanwise extent and the time persistence of uniform momentum zones (UMZs) in a zero pressure gradient turbulent boundary layer (ZPG-TBL) and a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation. The instantaneous detection methodology introduced by Adrian et al. [1] is used to detect the UMZs and is extended to take into account the spanwise domain length and the temporal evolution of the UMZs. The Reynolds number based on friction velocity (Reτ) ranges from 1176 to 1277 for the ZPG-TBL and from 1652 to 1745 for the self-similar APG-TBL within the domain of interest. For both the TBL cases, probability density functions (PDFs) of the number of UMZs are computed as a function of the streamwise extent, spanwise extent and time extent. For the ZPG-TBL, when the streamwise length of the domain is greater than or equal to 3 boundary layer thickness, the probability of finding 4 UMZs becomes almost negligible. For the APG-TBL, even when the streamwise domain length is taken as large as 1.3 boundary layer thickness, the probability of finding 4 UMZs is still significant. The spanwise extent of the UMZs is found to be shorter than their streamwise extent regardless of the pressure gradient in the flow. In the ZPG-TBL flow, the majority of the UMZs have a spanwise extent of the order of one-tenth of a boundary layer thickness while for the APG-TBL, it is found to be on the order of one-hundredth of a boundary layer thickness. In the ZPG-TBL, the probability of finding 2 UMZs that persist over a time period of 2 integral time scale is around 50%. Similarly, for the APG-TBL, the probability of finding 2 UMZs with a time persistence of 0.4 integral time scale is over 50%. In the case of the ZPG-TBL, it is observed that some of the UMZs with higher persistence in time have higher streamwise momentum and are found to be closer to the free-stream in general. This result is consistent with the previous observations by Laskari et al. [2]. In contrast, for the APG-TBL, UMZs with longer time persistence are found closer to the wall with lower streamwise momentum.

Asymptotic Estimate of Stability of a Supersonic Boundary Layer in a Vibrationally Excited Gas on a Plate

An asymptotic theory of the neutral stability curve of a supersonic boundary layer in a vibrationally excited molecular gas on a flat plate is constructed. The equations of the two-temperature viscous heat-conducting gas dynamics are considered as the initial mathematical model of the flow. On the basis of their linearization about the self-similar boundary layer solution for a perfect gas, a spectral problem is derived for the eighth-order system of linear ordinary differential equations. An algebraic secular equation with a typical decoupling into inviscid and viscous parts is derived from the linear combination of the boundary values of its solutions decreasing outside the boundary layer which solved numerically. It is shown that the neutral stability curves calculated in this way confirm the effect of increasing flow stability against the background of the relaxation process and within 12–15% agree with the previously obtained results of the direct numerical solution of the full spectral problem. The solution of the simplified system of equations for calculating the critical Reynolds number gives a similar result.

Self-similar compressible turbulent boundary layers with pressure gradients. Part 1. Direct numerical simulation and assessment of Morkovin’s hypothesis

A direct numerical simulation study of self-similar compressible flat-plate turbulent boundary layers (TBLs) with pressure gradients (PGs) has been performed for inflow Mach numbers of 0.5 and 2.0. All cases are computed with smooth PGs for both favourable and adverse PG distributions (FPG, APG) and thus are akin to experiments using a reflected-wave set-up. The equilibrium character allows for a systematic comparison between sub- and supersonic cases, enabling the isolation of pure PG effects from Mach-number effects and thus an investigation of the validity of common compressibility transformations for compressible PG TBLs. It turned out that the kinematic Rotta–Clauser parameter $\unicode[STIX]{x1D6FD}_{K}$ calculated using the incompressible form of the boundary-layer displacement thickness as length scale is the appropriate similarity parameter to compare both sub- and supersonic cases. Whereas the subsonic APG cases show trends known from incompressible flow, the interpretation of the supersonic PG cases is intricate. Both sub- and supersonic regions exist in the boundary layer, which counteract in their spatial evolution. The boundary-layer thickness $\unicode[STIX]{x1D6FF}_{99}$ and the skin-friction coefficient $c_{f}$, for instance, are therefore in a comparable range for all compressible APG cases. The evaluation of local non-dimensionalized total and turbulent shear stresses shows an almost identical behaviour for both sub- and supersonic cases characterized by similar $\unicode[STIX]{x1D6FD}_{K}$, which indicates the (approximate) validity of Morkovin’s scaling/hypothesis also for compressible PG TBLs. Likewise, the local non-dimensionalized distributions of the mean-flow pressure and the pressure fluctuations are virtually invariant to the local Mach number for same $\unicode[STIX]{x1D6FD}_{K}$-cases. In the inner layer, the van Driest transformation collapses compressible mean-flow data of the streamwise velocity component well into their nearly incompressible counterparts with the same $\unicode[STIX]{x1D6FD}_{K}$. However, noticeable differences can be observed in the wake region of the velocity profiles, depending on the strength of the PG. For both sub- and supersonic cases the recovery factor was found to be significantly decreased by APGs and increased by FPGs, but also to remain virtually constant in regions of approximated equilibrium.

Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer

A thorough self-similarity analysis is presented to investigate the properties of self-similarity for the outer layer of compressible turbulent boundary layers. The results are validated using the compressible and quasi-incompressible direct numerical simulation (DNS) data shown and discussed in the first part of this study; see Wenzelet al. (J. Fluid Mech., vol. 880, 2019, pp. 239–283). The analysis is carried out for a general set of characteristic scales, and conditions are derived which have to be fulfilled by these sets in case of self-similarity. To evaluate the main findings derived, four sets of characteristic scales are proposed and tested. These represent compressible extensions of the incompressible edge scaling, friction scaling, Zagarola–Smits scaling and a newly defined Rotta–Clauser scaling. Their scaling success is assessed by checking the collapse of flow-field profiles extracted at various streamwise positions, being normalized by the respective scales. For a good set of scales, most conditions derived in the analysis are fulfilled. As suggested by the data investigated, approximate self-similarity can be achieved for the mean-flow distributions of the velocity, mass flux and total enthalpy and the turbulent terms. Self-similarity thus can be stated to be achievable to a very high degree in the compressible regime. Revealed by the analysis and confirmed by the DNS data, this state is predicted by the compressible pressure-gradient boundary-layer growth parameter$\unicode[STIX]{x1D6EC}_{c}$, which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption,$\unicode[STIX]{x1D6EC}_{c}$values become comparable for compressible and incompressible pressure-gradient cases with similar wall-normal shear-stress distributions. The Rotta–Clauser parameter in its traditional form$\unicode[STIX]{x1D6FD}_{K}=(\unicode[STIX]{x1D6FF}_{K}^{\ast }/\bar{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$with the kinematic (incompressible) displacement thickness$\unicode[STIX]{x1D6FF}_{K}^{\ast }$is shown to be a valid parameter of the form$\unicode[STIX]{x1D6EC}_{c}$and hence still is a good indicator for equilibrium flow in the compressible regime at the finite Reynolds numbers considered. Furthermore, the analysis reveals that the often neglected derivative of the length scale,$\text{d}L_{0}/\text{d}x$, can be incorporated, which was found to have an important influence on the scaling success of common ‘low-Reynolds-number’ DNS data; this holds for both incompressible and compressible flow. Especially for the scaling of the$\bar{\unicode[STIX]{x1D70C}}\widetilde{u^{\prime \prime }v^{\prime \prime }}$stress and thus also the wall shear stress$\bar{\unicode[STIX]{x1D70F}}_{w}$, the inclusion of$\text{d}L_{0}/\text{d}x$leads to palpable improvements.

Volumetric measurements of a self-similar adverse pressure gradient turbulent boundary layer using single-camera light-field particle image velocimetry

As a novel volumetric particle image velocimetry technique, single-camera light-field PIV (LF-PIV) is able to acquire three-dimensional flow fields through a single camera. Compared with other multi-camera 3D PIV techniques, LF-PIV has distinct advantages, including concise hardware setup and low optical access requirements. Its capability has proven effective in many experimental investigations. In this study, the use of LF-PIV in measuring a self-similar adverse pressure-gradient turbulent boundary layer (APG-TBL) is demonstrated. Experiments are performed in a large water tunnel at the Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC), Monash University. Sets of 250 light-field PIV image pairs are captured covering both the inner and outer regions of the boundary layer. Instantaneous 3D velocity fields are reconstructed using a GPU accelerated density ray tracing multiplicative reconstruction technique (DRT-MART) and three-dimensional cross-correlation methods. The LF-PIV results are compared with two-dimensional PIV (2D-PIV) measurements of the same flow. Comparable accuracy to 2D-PIV is achieved for first- and second-order velocity statistics above approximately $$ y/\delta_{1} = 1 $$ .

First-mode-induced nonlinear breakdown in a hypersonic boundary layer

The mechanisms underlying transition of a Mach 6 flat plate boundary layer with adiabatic wall conditions are explored with Direct Numerical Simulations (DNS) using a high-order hybrid upwind-centered method. The forcing employed is obtained from linear stability theory (LST), which is used to identify Tollmien–Schlichting (T-S) or first-mode instability in the hypersonic boundary layer (HBL). At small amplitudes, the DNS displays linear growth, and matches those obtained from LST, including the spanwise structure of the most amplified oblique wave in the first-mode as well as the spreading angle of the turbulent wedge induced by the oblique wave under transitional scenarios. The main emphasis of the paper is on the nonlinear breakdown process, obtained by forcing the DNS at a higher amplitude so that the linear amplification region yields to nonlinear saturation and breakdown of the first-mode. The oblique waves are seen to first evolve into lambda-vortices with peak perturbation amplitudes around the generalized inflection point (GIP) of the HBL. The fundamental forcing frequency gives rise to progressive superharmonics: These first appear within the foot region of the lambda-vortices, and spread in the spanwise direction into the ring-shaped vortices that develop along the symmetry plane of the oblique waves. Transition and loss of symmetry is initiated upon saturation of the fundamental and harmonics. A modal analysis used to extract each frequency component yields further insight into streak-formation in these modes, resulting in localized regions of peak skin friction near the transition location. The near-wall boundary layer in the transitional zone displays coherent spanwise structures, which represent compressed regions of fluid and induce high gradients over the surface. Towards the end of the breakdown region, the flow displays a broadband spectrum, and the anticipated inertial sub-range appears inside the boundary layer, as a prelude to self-similar turbulent boundary layer evolution.

The granular Blasius problem

We consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for $x&lt;0$) but rough downstream (imposing a no-slip condition on $x&gt;0$), with a sharp transition at $x=0$. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number $I$; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The $\unicode[STIX]{x1D707}(I)$ rheology (Jop et al., Nature, vol. 441 (7094), 2006, pp. 727–730) asserts that $\text{d}\unicode[STIX]{x1D707}/\text{d}I=O(1/I^{2})$ as $I\rightarrow \infty$, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the $\unicode[STIX]{x1D707}(I)$ rheology, with different dependencies of $\unicode[STIX]{x1D707}(I)$ on $I$, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.

Linear Stability of the Boundary Layer of Relaxing Gas on a Plate

The development of inviscid and viscous two-dimensional subsonic disturbances in the supersonic flat-plate boundary layer of a vibrationally excited gas is investigated on the basis of the linear stability theory. The system of two-temperature gas dynamics which includes the Landau-Teller relaxation equation is used as the initial model. Undisturbed flow is described by the self-similar boundary-layer solution for a perfect gas. It is shown that in the inviscid approximation excitation decreases the maximum growth rate of the most unstable second mode by 10–12% as compared with an ideal gas. The neutral stability curves are calculated for the first and second most unstable modes at the Mach numbers M = 2.2, 4.5, and 4.8. For both modes the critical Reynolds numbers at maximum excitation are greater by 12–13% than the corresponding values for the perfect gas.

Viscous Slip MHD Flow over a Moving Sheet with an Arbitrary Surface Velocity

The magnetohydrodynamic (MHD) flow induced by a stretching or shrinking sheet under slip conditions is studied.Analytical solutions based on the boundary layer assumption are obtained in a closed form and can be applied to a flow configuration with any arbitrary velocity distributions. Seven typical sheet velocity profiles are employed as illustrating examples. The solutions to the slip MHD flow are derived from the general solution and discussed in detail. Different from self-similar boundary layer flows, the flows studied in this work have solutions in explicit analytical forms. However, the current flows require special mass transfer at the wall, which is determined by the moving velocity of the sheet. The effects of the slip parameter, the mass transfer at the wall, and the magnetic field on the flow are also demonstrated.

Entrainment model for fully-developed wind farms: Effects of atmospheric stability and an ideal limit for wind farm performance

While a theoretical limit has long been established for the performance of a single turbine, no corresponding upper bound exists for the power output from a large wind farm, making it difficult to evaluate the available potential for further performance gains. Here we build a model describing the essential features of a large array of turbines with arbitrary design and layout, by considering a fully-developed wind farm whose upper edge is bounded by a self-similar boundary layer. The exchanges between the wind farm, the overlaying boundary layer, and the outer flow are parameterized by means of the classical entrainment hypothesis. We obtain a concise expression for the wind farm's power density (corresponding to power output per unit planform area), as a function of three coefficients, which represent the array thrust and the turbulent exchanges at each of the two interfaces. We assess the performance of our model by comparing the predicted power density to field data, laboratory measurements and large-eddy simulations for the fully-developed regions of wind farms, finding good agreement. We extend our model to include the effect of atmospheric stability on power output, by using an established parameterization relating entrainment coefficients to local Froude numbers. Our predictions for power variation with atmospheric stability are in broad agreement with field measurements. To the best of our knowledge, this constitutes the first quantitative comparison between an atmospheric-stability-dependent model and field data. Finally, we consider an ideal limit for array operation, whereby turbines are designed to maximize momentum exchange with the overlying boundary layer. This enables us to obtain an upper bound for the performance of large wind farms, which we determine to be an order of magnitude larger than the output of contemporary turbine arrays.

An Unsteady Separated Stagnation-Point Flow Towards a Rigid Flat Plate

This paper discusses an unsteady separated stagnation-point flow of a viscous fluid over a flat plate covering the complete range of the unsteadiness parameter β in combination with the flow strength parameter a (&gt;0). Here, β varies from zero, Hiemenz's steady stagnation-point flow, to large β-limit, for which the governing boundary layer equation reduces to an approximate one in which the convective inertial effects are negligible. An important finding of this study is that the governing boundary layer equation conceives an analytic solution for the specific relation β = 2a. It is found that for a given value of β (≥0) the present flow problem always provides a unique attached flow solution (AFS), whereas for a negative value of β the self-similar boundary layer solution may or may not exist that depends completely on the values of a and β (&lt;0). If the solution exists, it may either be unique or dual or multiple in nature. According to the characteristic features of these solutions, they have been categorized into two classes—one which is AFS and the other is reverse flow solution (RFS). Another interesting finding of this analysis is the asymptotic solution which is more practical than the numerical solutions for large values of β (&gt;0) depending upon the values of a. A novel result which arises from the pressure distribution is that for a positive value of β the pressure is nonmonotonic along the stagnation-point streamline as there is a pressure minimum which moves toward the stagnation-point with an increasing value of β &gt; 0.