Streamwise Pressure Gradient Research Articles

Revisiting History Effects in Adverse-Pressure-Gradient Turbulent Boundary Layers

The goal of this study is to present a first step towards establishing criteria aimed at assessing whether a particular adverse-pressure-gradient (APG) turbulent boundary layer (TBL) can be considered well-behaved, i.e., whether it is independent of the inflow conditions and is exempt of numerical or experimental artifacts. To this end, we analyzed several high-quality datasets, including in-house numerical databases of APG TBLs developing over flat-plates and the suction side of a wing section, and five studies available in the literature. Due to the impact of the flow history on the particular state of the boundary layer, we developed three criteria of convergence to well-behaved conditions, to be used depending on the particular case under study. (i) In the first criterion, we develop empirical correlations defining the Re\U0001d703-evolution of the skin-friction coefficient and the shape factor in APG TBLs with constant values of the Clauser pressure-gradient parameter β = 1 and 2 (note that β = δ∗/τwdPe/dx, where δ∗ is the displacement thickness, τw the wall-shear stress and dPe/dx the streamwise pressure gradient). (ii) In the second one, we propose a predictive method to obtain the skin-friction curve corresponding to an APG TBL subjected to any streamwise evolution of β, based only on data from zero-pressure-gradient TBLs. (iii) The third method relies on the diagnostic-plot concept modified with the shape factor, which scales APG TBLs subjected to a wide range of pressure-gradient conditions. These three criteria allow to ensure the correct flow development of a particular TBL, and thus to separate history and pressure-gradient effects in the analysis.

Analysis of turbulent axisymmetric inner near wake

ABSTRACTAn analysis is presented describing the characteristics of mean velocity profile in the axisymmetric turbulent inner near-wake flow behind a body of revolution. The near wake is developing under zero streamwise pressure gradient and the upstream turbulent boundary layer is fully developed. It is shown that the boundary layer condition that exists at the trailing edge can be used to describe the mean velocity profiles in the inner near wake. It is shown that the logarithmic layer of the upstream turbulent boundary layer continues to be valid for some more distance in the near wake, and as the streamwise distance is increased, the logarithmic layer is slowly getting destroyed. It is also shown that the central line velocity exhibits a logarithmic behaviour for large streamwise distance. Results of the analysis have been validated using available experimental data.

Equilibrium Scour-Depth Prediction around Cylindrical Structures

Offshore gravity base foundations (GBFs) are often designed with complex geometries. Such structures interact with local hydrodynamics, creating an adverse pressure gradient that is responsible for flow and scour phenomena, including the bed shear stress amplification. In this study, a method is presented for predicting clear-water scour around cylindrical structures with nonuniform geometries under the force of a unidirectional current. The interaction of the flow field with the sediment around these complex structures is described in terms of nondimensional parameters that characterize the similitude of water-sediment movement. The paper presents insights into the influence the streamwise depth-averaged Euler number has on the equilibrium scour around uniform and nonuniform cylindrical structures. Here, the Euler number is based on the depth-averaged streamwise pressure gradient (calculated using potential flow theory), the mean flow velocity, and the fluid density. Following a dimensional analysis, the controlling parameters were found to be the Euler number, pile Reynolds number, Froude number, sediment mobility number, and nondimensional flow depth. Based on this finding, a new scour-prediction equation was developed. This new method shows good agreement with the database of scour depths acquired in this study (R2=0.91)(R2=0.91). Measurements of the equilibrium scour depth around nonuniform cylindrical structures were used to show the importance of the Euler number in the scour process. Finally, the importance of the remaining nondimensional quantities with respect to scour was also investigated in this study.

An experimental investigation of the supersonic turbulent boundary layer subjected to convex curvature

By employing particle image velocimetry, the impact of convex curvature on the supersonic turbulent boundary layer with inflow Mach number of 2.95 is experimentally investigated. The test models are carefully designed to have a zero-pressure-gradient convex boundary layer to isolate the influence of the streamwise pressure gradient while looking at the impact of convex curvature only. The results are compared with the favorable-pressure-gradient case (with pressure gradient parameter [Formula: see text]). For both of the zero-pressure-gradient and favorable-pressure-gradient convex boundary layers, the log law is found to be well preserved in the profile of streamwise velocity with inner scaling. Both of the mean streamwise velocity and the principal strain rate of the zero-pressure-gradient case collapse after scaling. By accumulating the streamwise velocity along the wall normal direction, the convex curvature is found to have the potential of diminishing the organized motion of the large scale vortices, which contributes to the stabilization of the turbulence.

Active and Natural Suction at Forward-Facing Steps for Delaying Laminar–Turbulent Transition

Two-dimensional, direct numerical simulations are used to study the impact of active and natural (passive) suction at forward-facing steps on laminar–turbulent transition. Suction is applied through a gap in front of a step that is located in a flat-plate boundary-layer flow without streamwise pressure gradient. A steady base flow is used with a Mach number of 0.6. Subsequently, Tollmien–Schlichting waves are introduced by suction and blowing at the wall, and their growth over the surface imperfection is evaluated by factors. The investigated step heights are in the range of one to two times the local displacement thickness of the smooth flat plate. Both sharp and rounded step corners are investigated. Cases with and without suction are compared with the smooth flat plate without suction based on factors according to the method. Thus, it is shown that suction is capable to compensate or even overcompensate the negative impact of different steps on laminar–turbulent transition. The work concludes with a configuration that allows natural (passive) suction in front of a forward-facing step. Here, a significant reduction of the factor compared to the smooth flat plate without suction is possible. Further estimations indicate that a net drag reduction may be possible as well.

Two-layer electrified pressure-driven flow in topographically structured channels

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

Entrance Aspect Ratio Effect on S-Duct Inlet Performance at High-Subsonic Flow

This paper reports the internal performance evaluation of S-duct diffusers with different entrance aspect ratios as part of a parametric investigation of a generic S-duct inlet. The generic S-duct diffusers studied had a rectangular entrance (aspect ratios of 1.5 and 2.0) transitioning S-duct diffuser in high-subsonic (Mach number > 0.8) flow. The test section was manufactured using rapid prototyping to facilitate the parametric investigation of the geometry. Streamwise static pressure and exit-plane total pressure were measured in a test-rig using surface pressure taps and a five-probe rotating rake, respectively. The baseline and a variant were simulated through computational fluid dynamics (CFD). The investigation indicated the presence of streamwise and circumferential pressure gradients leading to a three-dimensional flow in the S-duct diffuser and to distortion at the exit plane. The static pressure recovery increased for the diffuser with the higher aspect ratio. Total pressure losses and circumferential and radial distortions at the exit plane were higher than that of the podded nacelle type of inlet. An increase in the total pressure recovery was observed for the increase in the aspect ratio for the baseline area ratio (1.57) S-ducts, but without a clear trend for the other area ratio (1.8) ducts. The work represents the development of a database on the performance of a particular type of generic inlet. This database will be useful for predicting the performance of aero-engines and air vehicles in high-subsonic flight.

RANS simulation of high-Re turbulent flows using an immersed boundary method in conjunction with wall modeling

Abstract In this paper, an immersed boundary method, called the domain-free discretization (DFD) method, is developed to simulate the turbulent flows at high Reynolds numbers governed by the RANS equations. The strategy of DFD is that the discrete form of partial differential equations at an interior mesh node may involve some nodes outside the solution domain. Wall-modeling techniques, such as the adaptive wall function and the two-layer wall model, are adopted to alleviate the requirement on the near-wall mesh size. In the two-layer wall model, the effect of streamwise pressure gradient can be taken into account. The no-penetration condition and the wall shear stress prescribed by wall modeling, instead of the no-slip condition, are enforced at the immersed boundary. The normal and tangential velocity components at an exterior dependent node are evaluated via the linear extrapolation in the normal direction in conjunction with the no-penetration condition and the prescribed shear stress, respectively. The pressure at an exterior dependent node is obtained by solving the simplified momentum equation, and the temperature is calculated via the Crocoo-Busemann relation. By using analytical solutions, appropriate boundary conditions for the SST k − ω turbulence model equations are specified and then the turbulence variables at an exterior dependent node are determined. Various subsonic and transonic high-Re turbulent flows over stationary and moving bodies have been simulated to validate the present developed DFD method.

On the impact of adverse pressure gradient on the supersonic turbulent boundary layer

By employing the particle image velocimetry, the mean and turbulent characteristics of a Mach 2.95 turbulent boundary layer are experimentally investigated without the impact of curvature. The physical mechanism with which the streamwise adverse pressure gradient affects the supersonic boundary layer is revealed. The data are compared to that of the concave boundary layer with similar streamwise distributions of wall static pressure to clarify the separate impacts of the adverse pressure gradient and the concave curvature. The logarithmic law is observed to be well preserved for both of the cases. The dip below the logarithmic law is not observed in present investigation. Theoretical analysis indicates that it could be the result of compromise between the opposite impacts of the compression wave and the increased turbulent intensity. Compared to the zero pressure gradient boundary layer, the principal strain rate and the turbulent intensities are increased by the adverse pressure gradient. The shear layer formed due the hairpin packets could be sharpened by the compression wave, which leads to higher principal strain rate and the associated turbulent level. Due to the additional impact of the centrifugal instability brought by the concave wall, even higher turbulent intensities than that of the adverse pressure gradient case are introduced. The existence of velocity modes within the zero pressure gradient boundary layer suggests that the large scale motions are statistically well organized. The generation of new velocity modes due to the adverse pressure gradient indicates that the turbulent structure is changed by the adverse pressure gradient, through which more turbulence production that cannot be effectively predicted by the Reynolds-stress transport equations could be brought.

Convection induced by instabilities in the presence of a transverse seepage

The transition of laterally heated flows in a vertical layer and in the presence of a streamwise pressure gradient is examined numerically for the case of different values Prandtl number. The stability analysis of the basic flow for the pure hydrodynamic case (Pr=0) was reported in [1]. We find that in the absence of transverse pumping the previously known critical parameters are recovered [2], while as the strength of the Poiseuille flow component is increased the convective motion is delayed considerably. Following the linear stability analysis for the vertical channel flow our attention is focused on a study of the finite amplitude secondary travelling-wave (TW) solutions that develop from the perturbations of the transverse roll type imposed on the basic flow and temperature profiles. The linear stability of the secondary TWs against three-dimensional perturbations is also examined and it is shown that the bifurcating tertiary flows are phase-locked to the secondary TWs.

Effects of wall curvature and streamwise pressure gradient on film cooling effectiveness

As the load of a modern turbine is increasing, the turning angle and the flow acceleration is increasing. The effects of streamwise pressure gradient (SPG) and wall curvature on film cooling performance are becoming more and more important. Film cooling effectiveness with those two effects was investigated experimentally and numerically. Flow on the concave wall has an unstable effect and increases the turbulence intensity which leads to a faster dissipation and wider lateral spreading. With these two opposite effects, there is a critical point that when the momentum ratio is smaller than the certain value, film cooling effectiveness is highest on concave wall and the rank is opposite when the momentum ratio is larger than the value. The critical momentum ratio is 0.6 with nonaccelerated main flow for all density ratios. For the cases with different SPG, the critical momentum ratio still exists, while the value increases with stronger favorable pressure gradient and ranges from 0.55 to 0.75 in the present research. The film cooling performance with different SPG and wall curvature is quite different from the flat wall case and thus these effects should be taken into consideration in gas turbine film cooling design.

Pressure Gradient and Nonadiabatic Surface Effects on Boundary Layer Transition

The influence of the streamwise pressure gradient and a nonadiabatic surface on boundary layer transition was experimentally investigated at the Cryogenic Ludwieg-Tube Göttingen, Germany. Boundary layer transition was detected nonintrusively by means of the temperature-sensitive paint technique. The wind-tunnel model was designed to achieve a quasi-uniform streamwise pressure gradient over a large portion of the model chord length. This allowed the effects on boundary layer transition of the streamwise pressure gradient and wall temperature ratio to be decoupled. The model was tested at high Reynolds numbers and at a high subsonic Mach number. Favorable, almost-zero, and adverse streamwise pressure gradients were considered; and various temperature differences between the flow and the model surface were implemented. Stronger flow acceleration and lower wall temperature ratios led to an increase of the transition Reynolds number. Larger increases in the transition Reynolds number were obtained at more pronounced flow acceleration for the same reduction in the wall temperature ratio. The measured pressure distributions served as input for boundary layer stability computations, performed according to compressible, linear, local stability theory under the (quasi-) parallel-flow assumption. Amplification factors of the Tollmien–Schlichting waves were shown to be reduced by stronger favorable pressure gradients and lower wall temperature ratios, which was in agreement with the observed variation in the transition Reynolds number.

Design and Scaling of Plasma Streamwise Vortex Generators for Flow Separation Control

The design and scaling of plasma streamwise vortex generators in a variable pressure gradient environment are presented. This involves measurements in the boundary layer on the suction surface of a V-22 wing model which provided a generic flowfield that undergoes trailing-edge flow separation as a precursor to lift stall. Experiments were performed at freestream Mach numbers from 0.1 to 0.2, which resulted in chord Reynolds numbers of to . Different streamwise pressure gradients were examined by changing the airfoil angle of attack. Three components of the mean velocity in the boundary layer on the suction side of the wing model were measured using a five-hole pitot probe. These measurements were used to quantify the plasma streamwise vortex generator generated mean streamwise vorticity and the magnitude of the vorticity reorientation term in the vorticity transport equation that was used in the scaling analysis of the plasma streamwise vortex generator performance. The results validate the prediction of the streamwise vorticity production associated with the reorientation of the boundary-layer mean vorticity to scale as , where is the streamwise length of the plasma streamwise vortex generator array, is the external velocity at the crest of the airfoil, is the characteristic mean velocity of the flow over the plasma streamwise vortex generator array, is the change in the streamwise velocity produced by the plasma actuator, and is the spanwise spacing between the plasma streamwise vortex generator electrodes. The criteria for is that the convective timescale be greater than the timescale for vorticity reorientation in the boundary layer, namely, . Both the plasma streamwise vortex generator generated and were found to scale with . An optimally scaled plasma streamwise vortex generator array on the V-22 wing model produced as much as a 40 point drag count reduction at poststall angles of attack.

Laminar and turbulent flows over hydrophobic surfaces with shear-dependent slip length

Motivated by extensive discussion in the literature, by experimental evidence and by recent direct numerical simulations, we study flows over hydrophobic surfaces with shear-dependent slip lengths and we report their drag-reduction properties. The laminar channel-flow and pipe-flow solutions are derived and the effects of hydrophobicity are quantified by the decrease of the streamwise pressure gradient for constant mass flow rate and by the increase of the mass flow rate for constant streamwise pressure gradient. The nonlinear Lyapunov stability analysis, first applied to a two-dimensional channel flow by Balogh et al. [“Stability enhancement by boundary control in 2-D channel flow,” IEEE Trans. Autom. Control 46, 1696-1711 (2001)], is employed on the three-dimensional channel flow with walls featuring shear-dependent slip lengths. The feedback law extracted through the stability analysis is recognized for the first time to coincide with the slip-length model used to represent the hydrophobic surfaces, thereby providing a precise physical interpretation for the feedback law advanced by Balogh et al. The theoretical framework by Fukagata et al. [“A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces,” Phys. Fluids 18, 051703 (2006)] is employed to model the drag-reduction effect engendered by the shear-dependent slip-length surfaces and the theoretical drag-reduction values are in very good agreement with our direct numerical simulation data. The turbulent drag reduction is measured as a function of the hydrophobic-surface parameters and is found to be a function of the time- and space-averaged slip length, irrespective of the local and instantaneous slip behaviour at the wall. For slip parameters and flow conditions that could be realized in the laboratory, the maximum computed turbulent drag reduction is 50% and the drag reduction effect degrades when slip along the spanwise direction is considered. The power spent by the turbulent flow on the hydrophobic walls is computed for the first time and is found to be a non-negligible portion of the power saved through drag reduction, thereby recognizing the hydrophobic surfaces as a passive-absorbing drag-reduction method. The turbulent flow is further investigated through flow visualizations and statistics of the relevant quantities, such as vorticity and strain rates. When rescaled in drag-reduction viscous units, the streamwise vortices over the hydrophobic surface are strongly altered, while the low-speed streaks maintain their characteristic spanwise spacing. We finally show that the reduction of vortex stretching and enstrophy production is primarily caused by the eigenvectors of the strain rate tensor orienting perpendicularly to the vorticity vector.

Comparison of the Three-Dimensional Boundary Layer on Flat Versus Contoured Turbine Endwalls

The boundary layer on the endwall of an axial turbomachine passage is influenced by streamwise and cross-stream pressure gradients, as well as a large streamwise vortex, that develop in the passage. These influences distort the structure of the boundary layer and result in heat transfer and friction coefficients that differ significantly from simple two-dimensional boundary layers. Three-dimensional contouring of the endwall has been shown to reduce the strength of the large passage vortex and reduce endwall heat transfer, but the mechanisms of the reductions on the structure of the endwall boundary layer are not well understood. This study describes three-component measurements of mean and fluctuating velocities in the passage of a turbine blade obtained with a laser Doppler velocimeter (LDV). Friction coefficients obtained with the oil film interferometry (OFI) method were compared to measured heat transfer coefficients. In the passage, the strength of the large passage vortex was reduced with contouring. Regions where heat transfer was increased by endwall contouring corresponded to elevated turbulence levels compared to the flat endwall, but the variation in boundary layer skew across the passage was reduced with contouring.

Investigation on Secondary Flow Characteristics in a Curved Annular Duct with Struts

Under the influence of duct curvature, cross-sectional area variation and internal struts, the internal flow field within a curved annular duct becomes rather complicated and contains strong secondary flow. In this paper, the secondary flow characteristics in an annular duct with struts are experimentally and numerically investigated. The results show that large pressure gradients exist on the bends of hub and shroud. Meanwhile, two counter-rotating vortex pairs appear both along the hub-side and shroud-side surfaces. The hub-side vortex pair of which the vortex cores travel downstream parallelly evolves from the horseshoe vortex which is induced by the leading edge of the upstream strut, whereas the shroud-side vortex pair originates from the strut trailing edge and the corresponding vortex cores develop in a divergent way. Additionally, the effects of the duct exit Mach number on the secondary flow characteristics are also studied. As the exit Mach number increases, the streamwise pressure gradients increase and lead to more intense vortices, higher total pressure loss and larger flow distortion.

Asymptotic accelerated boundary layer over the permeable wall

The paper reports on numerical and analytical investigation of the laminar boundary layer with a favorable pressure gradient over a permeable wall. Researchers have obtained an analytical solution of boundary layer equations for asymptotic flow conditions. This solution allows proposing a relative asymptotic skin-friction function, which determines a degree of influence of flow acceleration and permeable wall on the flow. There are ranges of this function, where effects of permeable wall and streamwise pressure gradient have to be considered only in combination. Numerical simulation has showed that such combined influence of favorable pressure gradient and permeable wall extends the asymptotic flow. The study of a strong gas blowing into the accelerated flow has revealed that favorable pressure gradient impedes separation of the boundary layer. At that asymptotic flow starts from the point, where the separation would occur at gas injection into the zero-pressure gradient flow.

Study of laminar–turbulent flow transition under pulsatile conditions in a constricted channel

In this paper, direct numerical simulation is performed to investigate a pulsatile flow in a constricted channel to gain physical insights into laminar–turbulent–laminar flow transitions. An in-house computer code is used to conduct numerical simulations based on available high-performance shared memory parallel computing facilities. The Womersley number tested is fixed to 10.5 and the Reynolds number varies from 500 to 2000. The influences of the degree of stenosis and pulsatile conditions on flow transitions and structures are investigated. In the region upstream of the stenosis, the flow pattern is primarily laminar. Immediately after the stenosis, the flow recirculates under an adverse streamwise pressure gradient, and the flow pattern transitions from laminar to turbulent. In the region far downstream of the stenosis, the flow becomes re-laminarised. The physical characteristics of the flow field have been thoroughly analysed in terms of the mean streamwise velocity, turbulence kinetic energy, viscous wall shear stresses, wall pressure and turbulence kinetic energy spectra.

Experimental and Numerical Investigations of Shock-Film Cooling Interaction on a Turbine Blade With Fan-Shaped Cooling Holes

This paper presents the findings of an experimental and numerical investigation on the shock effect on heat transfer coefficient and film-cooling effectiveness. In this study, coolant was injected on the blade surface through a fan-shaped hole in a transonic cascade. The experimental results indicate that on the film-cooled suction surface of the blade, the shock from the adjacent blade impinging on the suction surface causes the film-cooling effectiveness to drop quickly by 18%, and then stay at a low level downstream of the shock. The shock also causes the local heat transfer coefficient to decrease rapidly by 25%, but then rise back up immediately after the shock. The results from the numerical study supported the film-cooling effectiveness and heat transfer coefficient trends that were observed in the experiment. A detailed analysis of the numerical results reveals that the rapid change of the film-cooling effectiveness is due to the near surface secondary flows, which push the hot mainstream air toward the injection centerline and lifts the low temperature core away from the surface. This secondary flow is a result of a spanwise pressure gradient. The drop in heat transfer coefficient is caused by a boundary layer separation bubble which results from an adverse streamwise pressure gradient at the shock position.

Numerical investigation of transition in a boundary layer subjected to favourable and adverse streamwise pressure gradients and elevated free stream turbulence

Laminar-to-turbulent transition of a boundary layer subjected to streamwise pressure gradients and elevated free stream turbulence is computed through direct numerical simulation. The streamwise pressure distribution and elevated free stream turbulence levels mimic the conditions present on the suction side of highly-cambered airfoils. Longitudinal streamwise streaks form in the laminar boundary layer through the selective inclusion of low-frequency disturbances from the free stream turbulence. The spanwise spacing normalized by local inner variables indicates stabilization of the streaks occurs by the favourable pressure gradient and prevents the development of secondary streak instability modes until downstream of the suction peak. Two distinct processes are found to trigger transition to turbulence in the adverse pressure gradient region of the flow. One involves the development of varicose secondary instability of individual low-speed streaks that results in their breakdown and the formation and growth of discrete turbulent spots. The other involves a rapid amplification of free stream disturbances in the inflectional boundary layer in the adverse pressure gradient region that results in a largely homogeneous breakdown to turbulence across the span. The effect of high-frequency free stream disturbances on the streak secondary instability and on the nonlinear processes within the growing turbulent spot are analysed through the inviscid transport of instantaneous vorticity. The results suggest that free stream turbulence contributes to the growth of the turbulent spot by generating large strain rates that activate vortex-stretching and tilting processes within the spot.