Favorable Pressure Gradient Region Research Articles

Implementation and validation of a generalized wall stress function

The generalized wall function by Shih et al. [Report No. M-1999-209398 (1999)], which accounts for non-equilibrium effects by the presence of favorable and adverse pressure gradients in turbulent flows, is addressed with the aim of performing high Reynolds number large-eddy simulations of the wall-bounded flow. The model uses a corrected law of the wall with a pressure gradient contribution to approximate the wall stress and applies to the entire viscous layer, buffer layer, and inertial region. A fully developed channel flow is first tested to validate the solver and model implementation, and then the wall function is assessed for the flow over a periodic hill. Wall-resolved simulations are in good agreement with reference results. A priori investigation with own experimental results corroborates the mathematical form of the model and suggests using different coefficients. The wall-modeled simulations show that the implemented wall model is able to improve the wall shear stress predictions compared to a standard equilibrium wall model. It corrects the underestimation of wall shear stresses by equilibrium models in the favorable pressure gradient region and the overestimation of wall shear stresses in the adverse pressure gradient region. The positions of the separation and reattachment points are also in good agreement with reference results. Furthermore, the prediction of the wall shear stress maximum in the favorable pressure gradient zone at the windward side of the hill is quite robust against coarsening the wall-normal grid spacing.

A streamline coordinate analysis of a turbulent boundary layer subject to pressure gradients and curvature on the windward side of a bump

Direct numerical simulation (DNS) of a turbulent boundary layer over the Gaussian (Boeing) bump is performed. This boundary layer exhibits a series of adverse and favourable pressure gradients and convex and concave curvature effects before separating. These effects on turbulent boundary layers are characterised and compared with a lower-Reynolds-number flow over the same geometry. The momentum budgets are analysed in the streamline-aligned coordinate system upstream of the separation region. These momentum budgets allow the simplification of equations to facilitate an integral analysis. Integral-analysis-based approximations for Reynolds stresses in the inner and outer regions of the boundary layer are also formulated. The shear and wall-normal Reynolds stress profiles normalised by these approximations exhibit a better collapse compared with friction velocity and Zagarola–Smits normalisations in the strong favourable pressure gradient region and in the mild adverse pressure region that precedes it in this flow. Simplification of these Reynolds stress approximations along with results from the DNS are used to obtain semi-empirical approximations that are able to provide stress closure in terms of wall solution fields for the turbulent boundary layer under consideration.

Comparison of smooth- and rough-wall non-equilibrium boundary layers with favourable and adverse pressure gradients

Measurements were made in rough-wall boundary layers subject to favourable, zero and adverse pressure gradients. Profiles of mean velocity and turbulence quantities were acquired and velocity fields were measured in multiple planes to document flow structure. Comparisons were made to equivalent smooth-wall cases with the same free stream velocity distributions. Outer layer similarity was observed between the rough- and smooth-wall cases in all quantities in the favourable and zero pressure gradient regions, but large differences were observed with adverse pressure gradients. In both the smooth- and rough-wall cases, the favourable pressure gradient reduced the turbulence in the boundary layer, and increased the size of turbulence structures relative to the boundary layer thickness in both the streamwise and spanwise directions, while lowering their inclination angle with respect to the wall. When the boundary layer was returned to a zero pressure gradient following the favourable pressure gradient region, the turbulence level and the size and inclination of the structures returned to their canonical zero pressure gradient condition. The response of the boundary layer was somewhat faster in the rough-wall case, causing it to reach equilibrium in a shorter streamwise distance after the changes in pressure gradient than in the smooth-wall case. The adverse pressure gradient increased turbulence levels relative to the wall friction velocity, reduced the size of turbulence structures relative to the boundary layer thickness and increased their inclination angle. The changes with the adverse pressure gradient were significantly larger with the rough wall than the smooth. The results suggest that similarity might be achieved with adverse pressure gradients if smooth- and rough-wall cases with the same Clauser pressure gradient parameter history are compared.

Turbulence Modeling in Hypersonic Turbulent Boundary Layers Subject to Convex Wall Curvature

Hypersonic boundary layers are crucial in aerospace applications such as hypersonic glide vehicles, rockets, and other advanced space vehicles. Hypersonic flows present unique transport phenomena, including nonnegligible flow compression/dilation, extra strain rates, and large momentum/thermal gradients. In this paper, the performance of three widely used turbulence models is compared, namely, the standard , the shear stress transport (SST) , and the Spalart–Allmaras (SA). Based on our turbulence modeling assessment plus the analysis of turbulent transport equation budgets over the experimental geometry from a previous study at a Mach number of 4.9, a moderate supremacy of SA over two equation models was found. To back our conclusions, previous experiments and direct numerical simulations at Mach numbers around 5 have been employed. Overall, the three considered models exhibited a consistent ability to predict first-order statistics both inside and outside the boundary layer. The SST variants were capable of describing the amplification of the constant shear layer induced by the presence of an adverse pressure gradient (APG). Furthermore, the SST model also replicated the second peak of turbulence production induced by the concave wall. There was a more aggressive distortion of the boundary layer by APG than by favorable pressure gradient (FPG) as compared with a zero pressure gradient (ZPG) boundary layer. A reasonable performance by Walz’s equation in the FPG region is also shown, whereas a notable lack of agreement is seen in the APG. Overall, one could argue for the SA model’s best compromise between accurate predictions, numerical stability, and mesh resolution insensitivity in the FPG and ZPG regions, particularly, in outer or integral boundary-layer parameters such as or . That being said, the two-equation models are far superior in terms of predicting near-wall parameters (such as or ) or their ability to accurately describe the physics of the hypersonic boundary layer for APG regions (for instance, outer-secondary peaks of turbulent kinetic energy production).

Adaptive-surrogate-based robust optimization of transonic natural laminar flow nacelle

Natural Laminar Flow (NLF) technology is very effective for reducing the skin friction drag of aircraft engine nacelle, but the aerodynamic performance of NLF nacelle is highly sensitive to uncertain working conditions. Therefore, it’s imperative to incorporate uncertainties into the design of NLF nacelle. In this study, for a robust optimization of NLF nacelle and for improving its efficiency, an adaptive-surrogate-based robust optimization strategy is established, which is an iterative optimization process where the surrogate model is updated to obtain the real Pareto front of multi-objective optimization problem. A case study is carried out to validate its feasibility and effectiveness. The results show that the optimization increases the favorable pressure gradient region and the volume ratio of the nacelle by increasing its lip radius and reducing its maximum diameter. And the aerodynamic robustness of the NLF nacelle is mainly determined by the lip radius, maximum diameter of nacelle and location of the maximum diameter. Compared to the initial nacelle, the optimized nacelle maintains a wide range of low drag and high laminar flow ratio in the disturbance space, which extends the average laminar flow region to 21.6% and facilitates a decrease of 1.98 counts in the average drag coefficient.

Balancing Laminar Extension and Wave Drag for Transonic Swept Wings

At a transonic condition, the design of a natural laminar flow (NLF) wing is challenging because the extension of the laminar flow needs to be finely balanced with the potential wave drag increase. To achieve this balance, it is proposed to unlock the wing sweep and introduce a three-dimensional (3-D) contour shock-control bump (SCB) in the optimization of the NLF infinite swept wing aiming at total drag reduction. A 3-D Reynolds-averaged Navier–Stokes flow solver is extended to incorporate laminar–turbulent transition prediction due to streamwise and crossflow instabilities. The flow solver is integrated in a gradient-based optimization framework. The transition criteria including streamwise and crossflow instabilities are coupled in the sensitivity calculation using a discrete-adjoint solver. Transonic design optimization of the sectional profile and wing sweep angle is first conducted at a Mach number of 0.78. The optimization managed to alleviate the shock wave, reducing the pressure drag while it failed to extend the laminar flow. Furthermore, a combined optimization of the wing with a parameterized SCB at the same condition is carried out. The optimized wing with the SCB features a long favorable pressure gradient region and a low-sweep angle, which allows for a large proportion of laminar flow without significant pressure drag. The shock wave is controlled by the 3-D bump, and the SCB has little effect on the upstream flow.

On the influence of biomimetic shark skin in dynamic flow separation

The effect of shark skin on the boundary-layer separation process under dynamic conditions (maneuvers) has been studied experimentally. We use a foil covered with biomimetic shark skin to explore how this type of surface impacts boundary-layer dynamics in both steady and accelerating conditions. The effect of denticles is assessed via particle image velocimetry in the wake. It is shown that dynamic conditions and small-scale disturbances can mitigate boundary-layer separation through instantaneous modification of the local pressure-gradient distribution. For instance, the region of favourable pressure gradient can be extended by accelerating the foil. The acceleration results in a thinner separated shear layer on the foil surface when compared to the steady reference case. This remarkable difference indicates that local roughness (introduced through for instance biomimetic shark skin) may trigger an interaction with relatively large-scale structures in the boundary layer for effective boundary-layer control during unsteady propulsion and maneuvering.

Simulation of a turbulent flow subjected to favorable and adverse pressure gradients

This paper reports the results from a direct numerical simulation of an initially turbulent boundary layer passing over a wall-mounted “speed bump” geometry. The speed bump, represented in the form of a Gaussian distribution profile, generates a favorable pressure gradient region over the upstream half of the geometry, followed by an adverse pressure gradient over the downstream half. The boundary layer approaching the bump undergoes strong acceleration in the favorable pressure gradient region before experiencing incipient or very weak separation within the adverse pressure gradient region. These types of flows have proved to be particularly challenging to predict using lower-fidelity simulation tools based on various turbulence modeling approaches and warrant the use of the highest fidelity simulation techniques. The present direct numerical simulation is performed using a flow solver developed exclusively for graphics processing units. Simulation results are utilized to examine the key phenomena present in the flowfield, such as relaminarization/stabilization in the strong acceleration region succeeded by retransition to turbulence near the onset of adverse pressure gradient, incipient/weak separation and development of internal layers, where the sense of streamwise pressure gradient changes at the foot, apex and tail of the bump.

Bypass transition in boundary layers subject to strong pressure gradient and curvature effects

This paper aims at characterizing the bypass transition in boundary layers subject to strong pressure gradient and curvature effects. A series of highly resolved large-eddy simulations of a high-pressure turbine vane are performed, and the primary focus is on the effects of free-stream turbulence (FST) states on transition mechanisms. The turbulent fluctuations that have convected from the inlet first interact with the blunt blade leading edge, forming vortical structures wrapping around the blade. For cases with relatively low-level FST, streamwise streaks are observed in the suction-side boundary layer, and the instabilities of the streaks cause the breakdown to turbulence. Moreover, the varicose mode of streak instability is predominant in the adverse pressure gradient region, while the sinuous mode is more common in the (weak) favourable pressure gradient region. On the other hand, for cases with higher levels of FST, the leading-edge structures are more irregularly distributed and no obvious streak instability is observed. Accordingly, the transition onset occurs much earlier, through the breakdown caused by interactions between vortical structures. Comparing between different cases, it is the competing effect between the FST intensity and the stabilizing pressure gradient that decides the path to transition and also the transition onset, whereas the integral length scale of FST affects the scales of the streamwise streaks in the boundary layer. Furthermore, while the streaks in the low-level FST cases are mainly induced by leading-edge vortical structures, the corresponding fluctuations show a stage of algebraic growth despite the weak favourable pressure gradient and curvature.

Influence of two-dimensional roughness element on boundary layer structure in the favourable pressure gradient region of the swept wing

In this article, the results of research about stationary and secondary disturbances development behind the localized and two-dimensional roughness elements are presented. It is shown that the two-dimensional roughness element has a destabilizing effect on the disturbances induced by the three-dimensional roughness element lying upstream. In this case, the two-dimensional roughness element causes the appearance of stationary structures, and then secondary perturbations, whose frequency range lies lower than in the case of the stationary vortices excited by a three-dimensional roughness element.

Large-eddy simulation of turbulent flow over a parametric set of bumps

Turbulent flow over a series of increasingly high, two-dimensional bumps is studied by well-resolved large-eddy simulation. The mean flow and Reynolds stresses for the lowest bump are in good agreement with experimental data. The flow encounters a favourable pressure gradient over the windward side of the bump, but does not relaminarize, as is evident from near-wall fluctuations. A patch of high turbulent kinetic energy forms in the lee of the bump and extends into the wake. It originates near the surface, before flow separation, and has a significant influence on flow development. The highest bumps create a small separation bubble, whereas flow over the lowest bump does not separate. The log law is absent over the entire bump, evidencing strong disequilibrium. This dataset was created for data-driven modelling. An optimization method is used to extract fields of variables that are used in turbulence closure models. From this, it is shown how these models fail to correctly predict the behaviour of these variables near to the surface. The discrepancies extend further away from the wall in the adverse pressure gradient and recovery regions than in the favourable pressure gradient region.

Structural Reynolds analogy in laminarescent boundary layers via DNS

Visualization of the thermal field in highly accelerated spatially developing turbulent boundary layers is carried out. Direct Numerical Simulation with high spatial/temporal resolution is performed in sink flow-type boundary layer by prescribing a passive scalar with a Prandtl number of 0.71. The range of the momentum thickness Reynolds number is approximately 320–432. The very strong Favorable Pressure Gradient (FPG) is imposed by a top converging shearless surface, which produces an approximately constant acceleration parameter of $$K = 4.0 \times 10^{-6}$$ . A precursor zone is prescribed upstream of the FPG region in order to generate accurate turbulent inflow information by means of the methodology proposed by Araya et al. (J Fluid Mech 670:518–605, 2011). While evident “signatures” of the very strong FPG have been identified in the velocity field, those “signatures” are much less evident in the temperature field causing a breakdown of the Reynolds analogy provoked by the streamwise pressure gradient (or source of dissimilarity between the momentum and thermal transport). A slow decrease in the thermal boundary layer thickness, $$\delta _T$$ , is observed in the FPG zone. Additionally, local maxima of absolute intensities of the thermal fluctuations $$t'_\mathrm{RMS}$$ exhibit nearly constant values (approx. 14% of $$T_{\infty }$$ ) along the composite domain. Conversely, absolute intensities of the streamwise ( $$u'_\mathrm{RMS}$$ ) show increases on their local maxima, with mild decreases in the transversal components of the velocity ( $$v'_\mathrm{RMS}$$ and $$w'_\mathrm{RMS}$$ ). Furthermore, profiles of cross-correlation $$ $$ at different streamwise stations depict a good collapsing level up to 4% of $$\delta$$ in the wall-normal direction, with peak values displacing farther from the wall in the FPG zone.

Actively Reduced Airfoil Drag by Transversal Surface Waves

The flow over a DRA2303 wing section at a Reynolds number of Re = 400,000 is actively controlled by spanwise traveling transversal surface waves. The actuated low-Mach number flow is investigated by a high-resolution large-eddy simulation. Approximately 74% of the solid surface on both sides of the wing section is deflected by a sinusoidal space- and time-dependent function in the wall-normal direction. The turbulence intensitites and wall-normal vorticity fluctuations are significantly reduced and a shift from one-dimensional turbulence to two-dimensional turbulence is observed. Besides a viscous drag reduction by 8.6% with a strong decrease of skin-friction in the favorable pressure gradient region and an overall drag decrease by 7.5%, a slight increase in lift is achieved for an external flow over a realistic geometry.

Shape optimization and experimental research of near space airship

Shape optimization has important effects on drag reduction of the near-space airship. This paper uses the Bezier curve to parameterize the hull of the airship. Based on multiple island genetic algorithms, the optimization platform combined with different programs is established, and a kind of low drag hull is obtained by optimization. Force measurement and flow observation wind tunnel test are used to research the aerodynamic characteristics of the ellipsoid hull and the optimized hull. Results show that, optimization mainly increases the volume ratio and the favorable pressure gradient region of the hull, therefore the surface area is reduced and transition position of the hull can be delayed. Compared with the LOTTE shape, transition position of the optimized shape moved backward by 13.78%, and the volume drag coefficient is reduced by 11.1%. It is known from the wind tunnel test that compared with the ellipsoid hull, transition position of the optimized shape moves backward obviously. Under the condition that the volume Reynolds number is 2.97 × 106, compared with the ellipsoid hull, volume drag coefficient of the optimized shape can reduce by 39.0%.

The logarithmic and power law behaviors of the accelerating, turbulent thermal boundary layer

Direct numerical simulation of spatially evolving thermal turbulent boundary layers with strong favorable pressure gradient (FPG) shows that the thermal fluctuation intensity, θ′+ and the Reynolds shear stress, u′v′¯+ exhibit a logarithmic behavior spanning the meso-layer (e.g., 50≤y+≤170). However, the mean thermal profile is not logarithmic even in the zero pressure gradient (ZPG) region; instead, it follows a power law. The maxima of u′2¯+ and v′θ′¯+ change little with the strength of acceleration, while v′+, w′+, and u′v′¯+ continue to decay in the flow direction. Furthermore, θ′+ and u′θ′¯+ surprisingly experience changes from constants in ZPG to sharp rises in the FPG region. Such behavior appears to be due to squashing of the streaks which decreases the streak flank angle below the critical value for “transient growth” generation of streamwise vortices, shutting down production [W. Schoppa and F. Hussain, “Coherent structure generation near-wall turbulence,” J. Fluid Mech. 453, 57–108 (2002)]. The streamwise vortices near the wall, although shrink because of stretching, simultaneously, also become weaker as the structures are progressively pushed farther down to the more viscous region near the wall. While the vortical structures decay rapidly in accelerating flows, the thermal field does not—nullifying the myth that both the thermal and velocity fields are similar.

Исследование структуры течения за двумерной шероховатостью на скользящем крыле в области благоприятного градиента давления

Исследование структуры течения за двумерной шероховатостью на скользящем крыле в области благоприятного градиента давления

Evolution of supersonic corner vortex in a hypersonic inlet/isolator model

There are complex corner vortex flows in a rectangular hypersonic inlet/isolator. The corner vortex propagates downstream and interacts with the shocks and expansion waves in the isolator repeatedly. The supersonic corner vortex in a generic hypersonic inlet/isolator model is theoretically and numerically analyzed at a freestream Mach number of 4.92. The cross-flow topology of the corner vortex flow is found to obey Zhang’s theory [“Analytical analysis of subsonic and supersonic vortex formation,” Acta Aerodyn. Sin. 13, 259–264 (1995)] strictly, except for the short process with the vortex core situated in a subsonic flow which is surrounded by a supersonic flow. In general, the evolution history of the corner vortex under the influence of the background waves in the hypersonic inlet/isolator model can be classified into two types, namely, from the adverse pressure gradient region to the favorable pressure gradient region and the reversed one. For type 1, the corner vortex is a one-celled vortex with the cross-sectional streamlines spiraling inwards at first. Then the Hopf bifurcation occurs and the streamlines in the outer part of the limit cycle switch to spiraling outwards, yielding a two-celled vortex. The limit cycle shrinks gradually and finally vanishes with the streamlines of the entire corner vortex spiraling outwards. For type 2, the cross-sectional streamlines of the corner vortex spiral outwards first. Then a stable limit cycle is formed, yielding a two-celled vortex. The short-lived limit cycle forces the streamlines in the corner vortex to change the spiraling trends rapidly. Although it is found in this paper that there are some defects on the theoretical proof of the limit cycle, Zhang’s theory is proven useful for the prediction and qualitative analysis of the complex corner vortex in a hypersonic inlet/isolator. In addition, three conservation laws inside the limit cycle are obtained.

Effect of mean and fluctuating pressure gradients on boundary layer turbulence

AbstractThis study focuses on the effects of mean (favourable) and large-scale fluctuating pressure gradients on boundary layer turbulence. Two-dimensional (2D) particle image velocimetry (PIV) measurements, some of which are time-resolved, have been performed upstream of and within a sink flow for two inlet Reynolds numbers, ${Re}_{\theta }(x_{1})=3360$ and 5285. The corresponding acceleration parameters, $K$, are ${1.3\times 10^{-6}}$ and ${0.6\times 10^{-6}}$. The time-resolved data at ${Re}_{\theta }(x_{1})=3360$ enables us to calculate the instantaneous pressure distributions by integrating the planar projection of the fluid material acceleration. As expected, all the locally normalized Reynolds stresses in the favourable pressure gradient (FPG) boundary layer are lower than those in the zero pressure gradient (ZPG) domain. However, the un-scaled stresses in the FPG region increase close to the wall and decay in the outer layer, indicating slow diffusion of near-wall turbulence into the outer region. Indeed, newly generated vortical structures remain confined to the near-wall region. An approximate analysis shows that this trend is caused by higher values of the streamwise and wall-normal gradients of mean streamwise velocity, combined with a slightly weaker strength of vortices in the FPG region. In both boundary layers, adverse pressure gradient fluctuations are mostly associated with sweeps, as the fluid approaching the wall decelerates. Conversely, FPG fluctuations are more likely to accompany ejections. In the ZPG boundary layer, loss of momentum near the wall during periods of strong large-scale adverse pressure gradient fluctuations and sweeps causes a phenomenon resembling local 3D flow separation. It is followed by a growing region of ejection. The flow deceleration before separation causes elevated near-wall small-scale turbulence, while high wall-normal momentum transfer occurs in the ejection region underneath the sweeps. In the FPG boundary layer, the instantaneous near-wall large-scale pressure gradient rarely becomes positive, as the pressure gradient fluctuations are weaker than the mean FPG. As a result, the separation-like phenomenon is markedly less pronounced and the sweeps do not show elevated small-scale turbulence and momentum transfer underneath them. In both boundary layers, periods of acceleration accompanying large-scale ejections involve near-wall spanwise contraction, and a high wall-normal momentum flux at all elevations. In the ZPG boundary layer, although some of the ejections are preceded, and presumably initiated, by regions of adverse pressure gradients and sweeps upstream, others are not. Conversely, in the FPG boundary layer, there is no evidence of sweeps or adverse pressure gradients immediately upstream of ejections. Apparently, the mechanisms initiating these ejections are either different from those involving large-scale sweeps or occur far upstream of the peak in FPG fluctuations.

Mechanisms for spatial steady three-dimensional disturbance growth in a non-parallel and separating boundary layer

Steady linear three-dimensional disturbances are investigated in a two-dimensional laminar boundary layer. The boundary layer is subject to a streamwise favourable-to-adverse pressure gradient and eventually undergoes separation. The separating flow corresponds to the first part of a pressure-induced laminar-separation bubble on a flat plate. Streamwise disturbance development in such a flow is studied by means of direct numerical simulation, a water-tunnel experiment and an adjoint-based parabolic theory suited to study spatial optimal growth. A complete overview of the disturbance evolution in various areas of the favourable-to-adverse pressure gradient laminar boundary layer is given. Results from all investigation methods show overall good agreement with respect to disturbance growth and shape within the entire domain. In the favourable pressure-gradient region and, again, slightly downstream of separation, transient growth caused by the lift-up effect dominates disturbance behaviour. In the adverse pressure-gradient region, a modal instability is observed. Evidence is presented that this instability is of Gortler type.

Transition mechanisms in a three-dimensional boundary-layer flow with pressure-gradient changeover

The laminar breakdown of a three-dimensional flat-plate boundary-layer flow with favourable and ensuing adverse pressure gradient (APG), generic for an infinite swept wing, is investigated in detail by means of spatial direct numerical simulations. Emphasis is on transition mechanisms in the region of adverse pressure gradient where the undistorted laminar base flow also becomes unstable with respect to Tollmien–Schlichting (TS) waves that are most amplified if two-dimensional with respect to the local streamwise direction. The influence of finite-amplitude crossflow vortices, coming from the region of favourable pressure gradient, on the TS instability properties is investigated. It turns out that crossflow-vortex-induced secondary instabilities are the most amplified disturbances even for low-amplitude vortex modes. The TS waves act as generators of fundamental low-frequency secondary modes, but are neither important for their growth nor for breakdown. As for active disturbance control, any method aiming at attenuating two-dimensional TS waves must fail. On the other hand, the passive upstream-flow-deformation technique also delays transition in the APG region.