Effects of the separating shear layer on the reattachment flow structure part 2: Reattachment length and wall shear stress

Measurements of reattachment length of a separated flow behind a backward-facing step for a range of Reynolds numbers (8000 < Re _H < 40,000) and initial boundary-layer thickness (0 < /H < 2) were performed with the purpose of explaining the scatter in existing (high quality) data sets and to understand the effect of the initial shear-layer structure on the reattachment zone. The reattachment length for the case of laminar boundary layers upstream of the step were 30% smaller than when the boundary layer upstream of the step was turbulent. Measured values of the mean wall shear stress in the reattachment zone were also measurably affected by the upstream boundary-layer state. The (rms) levels of fluctuating wall stress were not sensitive to boundary-layer state, but rather to /H, as was the case for the pressure profiles in part 1 (Adams and Johnston 1988).List of symbols C _* ^p normalized pressure, (C _p – C _{p, min})/(1 – C _{p, min}) - C _f skin friction coefficient, /0.5 U _ref ² - C _f level (rms) of fluctuation part of skin-friction coefficient - ER duct expansion ratio; outlet to inlet width - H step height - Re _d Reynolds number based on diameter - Re Reynolds number based on inlet boundary-layer momentum thickness, and U _ref - Re _H Reynolds number based on H and U _ref - x _r distance to reattachment - X ^* normalized distance, (x – x _r)/x _r (note: different from x/x _r in part 1) - /H ratio of inlet boundary-layer thickness to step height - gq ₀ momentum thickness upstream of step     

Upstream roughness and Reynolds number effects on turbulent flow structure over forward facing step

This paper presents results of experiments conducted to investigate the effects of Reynolds number and upstream wall roughness on the turbulence structure in the recirculation and recovery regions of a smooth forward facing step. A reference smooth upstream wall and a rough upstream wall made from sand gains were studied. For the smooth upstream wall, experiments were conducted at Reynolds number based on the freestream velocity and step height (h), Re_h = 4940, 8400 and 8650. The rough wall experiments was performed at Re_h = 5100, 8200 and 8600 to closely match the corresponding Re_h experiment over the smooth wall. The reattachment lengths in the smooth wall experiments were L_r/h ≈ 2.2, but upstream roughness significantly reduced these values to L_r/h ≈ 1.3. The integral scales within the recirculation bubbles were independent of upstream roughness and Reynolds number; however, upstream roughness significantly increased the spatial coherence and integral scales outside the recirculation bubbles and in the recovery region. Irrespective of the upstream wall condition, the redeveloping boundary layer recovered at 25h from reattachment.     

DNS and LES of Passively Controlled Turbulent Backward-Facing Step Flow

A passive control approach (no external energy input) for an unsteady separated flow case was investigated numerically. A surface-mounted control fence was positioned upstream of a backward-facing step, and as an oncoming flow a thin and fully developed turbulent boundary layer with a thickness of δ/h = 0.8 was used. The objective of the passive control was to enhance the entrainment rate of the shear layer bounding the separation zone behind the step, thereby reducing the mean reattachment length,〈 X _r0 〉. Direct Numerical Simulations (DNS) and Large-Eddy Simulations (LES) at Re _h = 3000 (based on the step height, h, and the free stream velocity, U _∞) were carried out for the uncontrolled and the controlled flow case. The LES results were in good agreement with the DNS reference solutions. Adaptively controlled feedback simulations showed that a certain minimum distance between the step edge and the upstream position of the control fence is required to achieve a maximum reduction of the reattachment length.     

Unsteady separated and reattaching turbulent flow over a two-dimensional square rib

The spatio-temporal characteristics of the separated and reattaching turbulent flow over a two-dimensional square rib were studied experimentally. Synchronized measurements of wall-pressure fluctuations and velocity fluctuations were made using a microphone array and a split-fiber film, respectively. Profiles of time-averaged streamwise velocity and wall-pressure fluctuations showed that the shear layer separated from the leading edge of the rib sweeps past the rib and directly reattaches on the bottom wall (x/H=9.75) downstream of the rib. A thin region of reverse flow was formed above the rib. The shedding large-scale vortical structures (fH/U₀=0.03) and the flapping separation bubble (fH/U₀=0.0075) could be discerned in the wall-pressure spectra. A multi-resolution analysis based on the maximum overlap discrete wavelet transform (MODWT) was performed to extract the intermittent events associated with the shedding large-scale vortical structures and the flapping separation bubble. The convective dynamics of the large-scale vortical structures were analyzed in terms of the autocorrelation of the continuous wavelet-transformed wall pressure, cross-correlation of the wall-pressure fluctuations, and the cross-correlation between the wall pressure at the time-averaged reattachment point and the streamwise velocity field. The convection speeds of the large-scale vortical structures before and after the reattachment point were U_c=0.35U₀ and 0.45U₀, respectively. The flapping motion of the separation bubble was analyzed in terms of the conditionally averaged reverse-flow intermittency near the wall region. The instantaneous reattachment point in response to the flapping motion was obtained; these findings established that the reattachment zone was a 1.2H-long region centered at x/H=9.75. The reverse-flow intermittency in one period of the flapping motion demonstrated that the thin reverse flow above the rib is influenced by the flapping motion of the separation bubble behind the rib.     

Visualization of Supersonic Low-Noise Flow Over Surface-Mounted Two-Dimensional Prisms

An experimental study of supersonic flow over two-dimensional surface-mounted prisms is carried out in a Mach 3 low-noise wind tunnel. The noise level of this supersonic wind tunnel, defined as the root mean-square Pitot pressure fluctuation normalized by the mean Pitot pressure, can be reduced to about 0.37%. The nanotracer planar laser scattering (NPLS) technique is used to analyze the influence of the prism geometry and the oncoming flow conditions on the typical flow structures including separation and reattachment shocks. With increase in the prism height the induced shocks move upstream. At a constant streamwise length L of a prism the timeaveraged NPLS images show that the length of the downstream recirculation region increases from 0.8L to 1.2L, when the prism height H changes from 3 to 5 mm. As compared with the flow structures occurring downstream of the prisms, the upstream flow structures are more susceptible to the oncoming boundary layer and are considerably different in laminar and turbulent flows. The separation shock wave is clearly visible in turbulent flow even for the 1-mm prism, whereas in the case of laminar flow there is no a distinct shock wave upstream of this prism. At the same time, the location of the flow reattachment and the angle of the reattachment shock wave in the downstream flow remain almost the same in both two flow regimes.     

Control of flow distortion in offset diffusers using trapped vorticity

Controlled concentrations of trapped vorticity within an offset, subsonic (M_AIP ≤ 0.7) diffuser are explored for active suppression of flow distortion in joint experimental and numerical investigations. The coupling between trapped vorticity, used to model boundary-layer separation, and secondary-flow vortices is manipulated using an array of fluidic oscillating jets, which are spanwise distributed just upstream of the trapped vortex. Actuation energizes the separated shear layer, reducing the size of separation and effecting an earlier reattachment of the boundary layer, which favorably effects the flow field downstream of reattachment. It is shown that optimal interactions between actuation and the trapped vortex fully suppress the central vortex pair, and redistributes the residual vorticity around the diffuser's circumference. This results in a 68% reduction in circumferential distortion at the Aerodynamic Interface Plane (AIP), using an actuation mass flow rate that is only 0.25% of the diffuser mass flow rate.     

Separation behaviour in front of a two-dimensional fence

The wall-bounded turbulent shear flow in front of a two-dimensional fence was investigated experimentally. In this prototype of a rapidly separating flow the assumptions for a first-order boundary-layer theory cease to apply. This is caused by the streamline curvature, the ensuing pressure gradient normal to the wall and the large vertical velocity component v in front of the fence. For the present experiment, where the ratio of the boundary layer thickness δ₀ measured without the fence and the fence height h is 0.82, the time mean separation length upstream of the fence l_f is 0.65h. However, instantaneous reverse flow events can be detected up to 4 mean separation lengths l_f upstream of the fence. The maximum value of the reverse flow factor χ_w is 95% indicating a strong reverse flow region. The experiments were performed by LDA and a wall pulsed-wire skin-friction meter. They show the limits of first-order boundary-layer theory and provide the first comprehensive data set of mean and fluctuating velocities and of wall shear-stress for this type of separating flow.     

Influence of the initial boundary layer on base pressure

The base pressure p_b, for an initial turbulent boundary layer, is determined for supersonic nonisothermal flow about a two-dimensional backward-facing step. This problem has been considered previously. In solving it in [1, 2], use was made of the Korst condition [3], which assumes equality of the total pressure p_j ^* on the line of constant mass to the pressure behind the closing oblique shock. However the pressure at the reattachment section p_* is lower than that behind the closing shock by 30–40% [4], and consequently the Korst condition is inaccurate. Therefore in the references cited only qualitative agreement with experiment was obtained. In contrast with [1, 3], Nash [5] introduces p_*; however, it is defined by an empirical coefficient. In the present study, to find p_b we make use of the condition of conservation of mass in the base region, written in the form of the equality p_j ^*=p_*, where p_* is defined from the assumption of minimum thickness of the dissipative layer at the reattachment section.Satisfactory agreement with the available experimental data is obtained without the use of correction factors. In the simplest case, when the thickness of the oncoming boundary layer ₁=0, the proposed method is no more complex than that of Korst. The determination of the base pressure with ₁=0 is considered in §1, and the determination with ₁>0 is considered in §2.     

Turbulent flow over a rough backward-facing step

This work characterizes the impacts of the realistic roughness due to deposition of foreign materials on the turbulent flows at surface transition from elevated rough-wall to smooth-wall. High resolution PIV measurements were performed in the streamwise-wall-normal (x–y) planes at two different spanwise positions in both smooth and rough backward-facing step flows. The experiment conditions were set at a Reynolds number of 3450 based on the free stream velocity U_∞ and the mean step height h, expansion ratio of 1.01, and the ratio of incoming boundary layer thickness to the step height, δ/h, of 8. The mean flow structures are observed to be modified by the roughness and they illustrate three-dimensional features in rough backward-facing step flows. The mean reattachment length X_r is significantly reduced by the roughness at one PIV measurement position while is slightly increased by the different roughness topography at the other measurement position. The mean velocity profiles at the reattachment point indicate that the studied roughness weakens the perturbation of the step to the incoming turbulent flow. Comparisons of Reynolds normal and shear stresses, productions of normal stresses, quadrant analysis of the instantaneous shear-stress contributing events, and mean spanwise vorticity reveal that the turbulence in the separated shear layer is reduced by the studied roughness. The results also indicate an earlier separation of the turbulent boundary layer over the current rough step, probably due to the adverse pressure gradient produced by the roughness topography even before the step.     

Reynolds number effects on a turbulent boundary layer with separation,reattachment, and recovery

The present paper addresses experimental studies of Reynolds number effects on a turbulent boundary layer with separation, reattachment, and recovery. A momentum thickness Reynolds number varies from 1,100 to 20,100 with a wind tunnel enclosed in a pressure vessel by varying the air density and wind tunnel speed. A custom-built, high-resolution laser Doppler anemometer provides fully resolved turbulence measurements over the full Reynolds number range. The experiments show that the mean flow is at most a very weak function of Reynolds number while turbulence quantities strongly depend on Reynolds number. Roller vortices are generated in the separated shear layer caused by the Kelvin–Helmholtz instability. Empirical Reynolds number scalings for the mean velocity and Reynolds stresses are proposed for the upstream boundary layer, the separated region, and the recovery region. The inflectional instability plays a critical role in the scaling in the separated region. The near-wall flow recovers quickly downstream of reattachment even if the outer layer is far from an equilibrium state. As a result, a stress equilibrium layer where a flat-plate boundary layer scaling is valid develops in the recovery region and grows outward moving downstream.     

A note on a boundary condition for spouted beds

The boundary condition, zero solids pressure at the top of a particle bed of maximum spoutable height, Hm, is shown to eliminate any resort to empiricism in the derivation of the fluid velocity in the annulus of a spouted bed for which both viscous and inertial effects are taken into account. The same boundary condition fails when applied to a spouted bed for which the bed height H 〈 Hm, especially when H 〈 0.8Hm.     

Experimental study of wall pressure fluctuations in rigid and elastic pipes behind an axisymmetric narrowing

Wall pressure fluctuations, p_t, in rigid and elastic pipes behind a local axisymmetric narrowing are studied. A sharp increase in their rms level in a finite region immediately downstream of the narrowing, leading up to a pronounced maximum upstream of the point of jet reattachment, is found. Approximate estimates both for the distance from the narrowing to the point of maximum rms pressure and for the rms magnitude at this point are obtained. Inspection of the wall pressure power spectrum, P, reveals the presence of low-frequency maxima. The maxima are found to be associated with the large-scale eddies in the regions of separated and reattached flow, and their frequencies are close to the characteristic frequencies of the eddies’ formation. These maxima are the main distinguishing features of the spectrum under investigation compared to the power spectrum of the wall pressure fluctuations in a fully-developed turbulent flow in a pipe without narrowing. A comparative analysis of the data for rigid and elastic pipes shows that changes in the pipe wall bending stiffness cause alterations in the flow structure near the wall and the corresponding redistribution of flow energy among the vortices. This results in an increase in the wall pressure amplitude and the low-frequency level of the wall pressure power spectrum, as well as the appearance of new frequency components in this domain.     

Visualization of a locally-forced separated flow over a backward-facing step

A laboratory water channel experiment was made of the separated flow over a backward-facing step. The flow was excited by a sinusoidally oscillating jet issuing from a separation line. The slit was connected to a cavity in which water was forced through a rigid pipe by a scotch-yoke system. The Reynolds number based on the step height (H) was fixed at Re _H =1200. The forcing frequency was varied in the range 0.305?St _H ?0.955 at the forcing amplitude A ₀=0.3. Time-averaged flow measurements were made by a LDV system, especially in the recirculating region behind the backward-facing step. To characterize the large-scale vortex evolution due to the local forcing, flow visualizations were performed by a dye tracer method with fluorescent ink. The vortex amalgamation process was captured at the effective forcing frequency (St _H =0.477) for laminar separation. This vortex merging process enhances flow mixing, which leads to the shortening of the reattachment length.     

Turbulent wall pressure reduction using suction control

A study of the fluctuating wall pressure beneath a 2-d turbulent boundary layer was conducted in a water tunnel with Reynolds numbers, based on momentum thickness, ranging between 2,100 and 4,300. The boundary layer was perturbed with steady mild suction to assess the effect of upstream suction on the fluctuating wall pressure measured downstream of the suction slit. Wall pressure signatures were captured using a custom-fabricated piezo-ceramic array with d ⁺ values ranging between 64 and 107. Likewise, the velocity field was measured with a laser Doppler velocimeter with l ⁺ values ranging between 4.0 and 6.7 for the lowest and highest Re _θ investigated. Estimates of the wall pressure spectra revealed a noticeable hydrodynamic peak that scaled reasonably well with outer variables and with an average convective speed of 75 % of the free stream velocity (based on unconditionally sampled pressure time series). Two boundary layer suction control cases were studied corresponding to suction rates of less then 30 % of the boundary layer momentum. The findings reveal how only modest amounts of suction are needed to reduce upwards 50–60 % of the hydrodynamic ridge.     

Large-scale spanwise periodicity in a turbulent boundary layer induced by highly ordered and directional surface roughness

The effect of converging–diverging riblet-type surface roughness (riblets arranged in a ‘herringbone’ pattern) are investigated experimentally in a zero pressure gradient turbulent boundary layer. For this initial parametric investigation three different parameters of the surface roughness are analysed in detail; the converging–diverging riblet yaw angle α, the streamwise fetch or development length over the rough surface F_x and the viscous-scaled riblet height h⁺. It is observed that this highly directional surface roughness pattern induces a large-scale spanwise periodicity onto the boundary layer, resulting in a pronounced spanwise modification of the boundary layer thickness. Hot-wire measurements reveal that above the diverging region, the local mean velocity increases while the turbulent intensity decreases, resulting in a thinner overall boundary layer thickness in these locations. The opposite situation occurs over the converging region, where the local mean velocity is decreased and the turbulent intensity increases, producing a locally thicker boundary layer. Increasing the converging–diverging angle or the viscous-scaled riblet height results in stronger spanwise perturbations. For the strongest convergent–divergent angle, the spanwise variation of the boundary layer thickness between the diverging and converging region is almost a factor of two. Such a large variation is remarkable considering that the riblet height is only 1% of the unperturbed boundary layer thickness. Increasing the fetch seems to cause the perturbations to grow further from the surface, while the overall strength of the induced high and low speed regions remain relatively unaltered. Further analysis of the pre-multiplied energy spectra suggests that the surface roughness has modified or redistributed the largest scale energetic structures.     

On the logarithmic-law constants and the turbulent boundary layer at low Reynolds numbers

It is shown that a family of formally derived similarity solutions describe to leading order the outer region of a turbulent boundary layer for all Reynolds numbers for which the layer satisfies the logarithmic law-of-the-wall. The family includes Coles' [1] hypothesis. For consistency with this hypothesis and the logarithmic law-of-the-wall, it is further shown that the constants in the latter form the product κC=2+O(ε), suggesting the logarithmic law of the wall be written $${U \mathord{\left/ {\vphantom {U {U_\tau = \kappa ^{ - 1} }}} \right. \kern-\nulldelimiterspace} {U_\tau = \kappa ^{ - 1} }}\ln \left( {e^2 U_\tau {y \mathord{\left/ {\vphantom {y \nu }} \right. \kern-\nulldelimiterspace} \nu }} \right) + O\left( \in \right).$$ A range of data are reprocessed to determine the skin friction coefficientC _f using κC = 2 and these collapse well when plotted against momentum thickness Reynolds number, Re _θ . It is also shown that the form parameter, Π, in Coles hypothesis is not unique but is determined by history effects peculiar to the boundary layer. Expressions are derived forC _f (Re _θ ) and the shape factorH (Re _θ ); both agree closely with the data and are valid over all Reynolds numbers for which the logarithmic law of the wall is satisfied.     

Unsteady near wake of a flat disk normal to a wall

The unsteady wake of a flat disk (diameter D) located at a distance of H from a flat plate has been experimentally investigated at a Reynolds number Re _D  = 1.3 × 10⁵. Tests have been performed for a range of gap ratio (H/D), spanning from 0.3 to 1.75. The leading edge of the flat plate is either streamlined (elliptical) or blunt (square). These configurations have been studied with PIV, high speed PIV and multi-arrayed off-set fluctuating pressure measurements. The results show a progressive increase of the complexity of the flow and of the interaction as the gap ratio decreases. For large values of H/D (1.75), the interaction is weak and the power spectral densities (PSD) exhibit a strong peak associated with the vortex shedding events (St = 0.131) – St = fD/U _∞ is the Strouhal number. For lower values of H/D (0.75), the magnitude of the wall fluctuating pressure increases significantly. A large band contribution is associated with the unsteady wake structure and turbulence. A slight increase of the shedding frequency (St = 0.145) is observed. A critical value of the gap ratio (about 0.35) has been determined. Below this critical value, a three-dimensional separated region is observed and the natural vortex shedding process is very strongly altered. These changes induce a great modification of the fluctuating pressure at the wall. Each interaction reacts in a different way to perturbed upstream conditions. In particular, the disk is an overwhelming perturbation for the lowest H/D value studied here and the relative influence of the upstream turbulence on the wall fluctuating pressure below the near wake region is moderate.     

Flow visualization of supersonic laminar flow over a backward-facing step via NPLS

An experimental study on a supersonic laminar flow over a backward-facing step of 5 mm height was undertaken in a low-noise indraft wind tunnel. To investigate the fine structures of Ma = 3.0 and 3.8 laminar flow over a backward-facing step, nanotracer planar laser scattering was adopted for flow visualization. Flow structures, including supersonic laminar boundary layer, separation, reattachment, redeveloping turbulent boundary layer, expansion wave fan and reattachment shock, were revealed in the transient flow fields. In the Ma = 3.0 BFS (backward-facing step) flow, by measuring four typical regions, it could be found that the emergence of weak shock waves was related to the K–H (Kelvin–Helmholtz) vortex which appeared in the free shear layer and that the convergence of these waves into a reattachment shock was distinct. Based on large numbers of measurements, the structure of time-averaging flow field could be gained. Reattachment occurred at the location downstream from the step, about 7–7.5 h distance. After reattachment, the recovery boundary layer developed into turbulence quickly and its thickness increased at an angle of 4.6°. At the location of X = 14h, the redeveloping boundary layer was about ten times thicker than its original thickness, but it still had not changed into fully developed turbulence. However, in the Ma = 3.8 flow, the emergence of weak shock waves could be seen seldom, due to the decrease of expansion. The reattachment point was thought to be near X = 15h according to the averaging result. The reattachment shock was not legible, which meant the expansion and compression effects were not intensive.     

Large eddy simulation of a developing turbulent boundary layer at a low Reynolds number

A spectral code has been used to simulate a developing turbulent boundary layer at low Reynolds number Re_θ (based on free stream velocity and momentum thickness) between 353 and 576. The starting field was generated by allowing a step change of temperature to diffuse outwards from one wall in a fully developed channel flow. The thermal boundary layer so created was conditionally sampled to convert it into a momentum boundary layer with an irrotational free stream region, a process which is justified by appeal to experiments. This initial field was allowed to develop until the momentum boundary layer thickness δ₉₉₅ had grown to about 1·5 times its original thickness. The results of the simulation have been compared with a wide range of experimental data. The outcome of this comparison is generally very satisfactory; the main trends of the experiments are well reproduced and our simulation supplements and extends the existing sets of experimental data. The simulation also gives pressure statistics which cannot be obtained experimentally. In particular, it gives the contribution of pressure diffusion to the balance equations for the Reynolds stress and indicates the error produced by omitting this term.     

Characteristics of the free-end mean pressure distribution for a surface-mounted finite-height cylinder

Wind tunnel experiments at a Reynolds number of Re = 6.5 × 10⁴ were used to study the effect of aspect ratio and boundary layer thickness on the mean static pressure distribution on the free end of a surface-mounted finite-height cylinder. The cylinder's aspect ratio was changed in small increments from AR = 0.5 to AR = 11. Two different boundary layer thicknesses (relative to the cylinder diameter) were employed, δ/D = 0.6 and δ/D = 1.9. From analysis of the mean pressure contour plots, it was found that the sizes and locations of regions of lower pressure, adverse pressure gradient, and higher pressure, and the appearance of “eye-like spots”, are sensitive to both AR and δ/D. The adverse pressure gradient occurs just ahead of the mean reattachment line while the eye-like spots are related to termination points of the legs of the arch vortex within the free-end mean recirculation zone. The total normal force coefficient experienced by the cylinder is strongly influenced by the contribution of the wall shear stress on the sides of the cylinder, with a change in direction of the net vertical shear stress contribution occurring between AR = 7 and AR = 8 for both boundary layers.