Spatial optimal growth in three-dimensional compressible boundary layers

Pulsating flow in a human aortic arch is studied from its kinematic and dynamic characteristics of transient tubular boundary layer. The results can only be obtained by a 3D fluid dynamic (CFD) analysis for the rapidly accelerated and decelerated systolic flow. The flow is based on a prescribed inlet velocity, V_O(t), which can be expressed as the instantaneous Reynolds number, <b>Re</b>(t) = ρDV_O/μ in which D is the tube diameter, ρ the blood density and μ the dynamic viscosity. Computation of pressure field requires a reference pressure at the downstream end section. The pressure is based on the pulse in Professor Y. C. Fung’s book on Biomechanics: Circulation. The finite volume method of an arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier-Stokes equations developed by Hirt et al. is used [2]. <br/> The results in the figure include pressure contours on the symmetric plane for <b>Re</b> = 12097, the longitudinal velocity and pressure distributions on a cross section ф = 113°. The peak systole is developed from t = 0.15 to 0.26 sec. At t = 0.18 sec. the instantaneous <b>Re</b> reaches 5104; the velocity and pressure are dominated by the rapid boundary layer development from the onset systole. The curvature effect is coupled with flow acceleration for spatial and temporal growth of boundary layer. The longitudinal velocity distribution developed from the inlet to section ф = 113° shows higher velocity occurred along the inner bend. The growth of tubular boundary layer is associated with a transient irrotational core. Similar phenomena appear in the upstream and downstream sections for spatial growth of boundary layer in early systole. The pressure distribution at the section shows higher pressure near the outer bend due to centrifugal forces. The effect is visible in pressure distribution on the longitudinal mid plane. The results indicate a rapid rising of pressure at t = 0.18 second with the predominant pressure drops toward the downstream. They are associated with acceleration during early systole. The pressure distribution on the symmetric plane for the peak flow at t = 0.26 second with <b>Re</b> reaching 12097 indicates insignificant longitudinal pressure gradients because of dVO/dt vanished momentary. At this instant, the presence of radial pressure variations along the curved segment is due to strong centrifugal forces. The w-contour becomes complicated along the inner bend characterized by a hump (refer to w = 75 cm/sec) with a pair of peak velocities (see w = 140 and 143.4 cm/sec) by its sides. The growth of three-dimensional boundary layers is associated with the secondary flow and pressure variations on the section. The pulsating flow continues with a rapid deceleration. Further development of laminar layer is demonstrated for flow when <b>Re</b> reduces to 5482. For the 3D pulsating flow, using the contour of w = 0 to demarcate backflow from forward stream is good for analyzing the 3D vortex. Also shown in Fig. 1 is the w-contour and pressure distribution for back flow when <b>Re</b> = -2132. The negative velocity (see w = -60 cm/sec) on the inner bend is in the laminar layer while the forward momentums remain on the outer bend. <br/> TKH first met Professor Fung at the 1970 Urodynamics Symposium where he presented a generalized Bernoulli’s equation with energy pumped from the ureter to urine bolus transport [3]. One professor from aerospace engineering made a comment: “I hate to attack a junior colleague. On God’s green world, there is no such Bernoulli equation”. At the conclusion of the conference, Professor Fung was asked by Dr. Saul Boyarsky for a comment. He said: <br/> “We are from the conventional engineering fields to work on unconventional biological system. Dr. Hung’s Bernoulli equation was correct”. <br/> This junior colleague, TKH, would later have the honor and opportunity to participate in many activities with Professor Fung, including ASME, ASCE, USNBC, ICMMB, JMMB, and NCKU.His admiration of Dr. Fung's kindness and friendship is evident in celebrating Professor Fung's 65th, 90th and 100th birthday conferences at UCSD. This paper is dedicated to Professor Yuan-Cheng Fung for his creativity and leadership in modern biomechanics and biomedical engineering.

The stability of an incompressible flat plate boundary layer with pressure gradient (Hartree pressure gradient parameter β = 1.0, 0.6, 0.4, 0.2, 0, −0.1, −0.14, and −0.1988) is computed from the linearized complete small disturbance equations. The analysis is nevertheless a quasiparallel treatment because although boundary layer growth is accounted for, the disturbance wave function is correct only in a strictly parallel flow. It is found that the nonparallel flow effect has a negligible influence on the critical Reynolds number Rδ*−c for 0.4 ≤ β ≤ 1.0, but leads to a decrease in Rδ*−c in the range −0.1988 ≤ β ≤ 0.4. The V terms lead to an increase in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to a decrease in the range −0.1988 ≤ β ≤ 0.4. The stream tube stretching term led to a decrease in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to an increase in the range 0.4 ≥ β ≥ −0.1988. The effect of the V terms and the stream tube stretching term ∂2U/∂x∂y appear to dominate all other boundary layer growth terms; near Rδ*−c for β = −0.1988 however, the stream tube stretching term and ∂2V/∂x2 appear to be of the same order of magnitude.

Stabilization of a Mach 5.92 Boundary Layer by Two-Dimensional Finite-Height Roughness

The effect of wall curvature on the full-coverage film cooling performance has been studied experimentally by using concave, flat and convex walls. The effect of coolant mass flux has also been investigated. A preliminary consideration, based on the balance of the centrifugal force associated with an injected fluid element and the pressure gradient normal to the wall, leads to the understanding that the trajectory of the injected fluid tends to move away from the concave wall and attach to the convex wall, respectively, under the present experimental conditions. This has been experimentally confirmed by both the boundary layer measurements and the smoke-wire flow-visualization. In accordance with the flow structure observed, the cooling effectiveness measured on the convex and concave walls is higher and lower than that on the flat wall. In addition, the mass flux ratio has little effect on the cooling effectiveness on the concave wall, while it is optimum in the range of 0.4 - 0.5 on the convex wall.

Electrodeposition of zinc on a carbon cathode followed by laser interferometry: Evaluation of the growth of the cathodic boundary layer

Spanwise wall oscillation is an easily implemented active control scheme. It can delay the transition to turbulent flows in incompressible boundary layers under appropriate control parameters. However, this delayed transition in compressible boundary layers has not yet been confirmed numerically. In this paper, we perform a series of direct numerical simulations of a spatially developing subsonic boundary layer with different spanwise wall oscillation periods and amplitudes. We numerically confirm the delay in the transition under appropriate control parameters. The delayed transition can be explained by the Stokes boundary layer induced by the spanwise wall oscillation affecting the streamwise vortices. The superposed spanwise velocity and streamwise vorticity observed in the Stokes boundary layer can weaken the development of instability in the original streamwise vortices under appropriate control parameters, leading to a delayed transition. Furthermore, the spanwise wall oscillation changes the flow compressibility, as can be inferred from the equation for the velocity divergence. The enhanced compressibility around the pre-transition regions also stabilizes the development of the streamwise vortices and causes a delayed transition.

The nonlinear evolution of a finite-amplitude disturbance in a 3-D supersonic boundary layer over a cone was investigated recently by Liu et al. using direct numerical simulation (DNS). It was found that certain small-scale 3-D disturbances amplified rapidly. These disturbances exhibit the characteristics of second modes, and the most amplified components have a well-defined spanwise wavelength, indicating a clear selectivity of the amplification. In the case of a cone, the three-dimensionality of the base flow and the disturbances themselves may be responsible for the rapid amplification. In order to ascertain which of these two effects are essential, in this study we carried out DNS of the nonlinear evolution of a spanwise localized disturbance (wavepacket) in a flat-plate boundary layer. A similar amplification of small-scale disturbances was observed, suggesting that the direct reason for the rapid amplification is the three-dimensionality of the disturbances rather than the three-dimensional nature of the base flow, even though the latter does alter the spanwise distribution of the disturbance. The rapid growth of 3-D waves may be attributed to the secondary instability mechanism. Further simulations were performed for a wavepacket of first modes in a supersonic boundary layer and of Tollmien-Schlichting (T-S) waves in an incompressible boundary layer. The results show that the amplifying components are in the band centered at zero spanwise wavenumber rather than at a finite spanwise wavenumber. It is therefore concluded that the rapid growth of 3-D disturbances in a band centered at a preferred large spanwise wavenumber is the main characteristic of nonlinear evolution of second mode disturbances in supersonic boundary layers.

Measurements have been performed of mean and fluctuating velocities as well as of Reynolds and wall shear stress in an incompressible axisymmetric turbulent boundary layer with a severe adverse pressure gradient. In the flow region where separation was incipient and, further downstream, where reverse flow occurred, only the x-components u and u′ and the skin friction were measured. The measurements were carried out by using conventional hot wires, pulsed wires and surface fences. A comparison of hot-wire and pulsed-wire measurements showed good agreement also at high local turbulence levels as long as the flow did not reverse; large discrepancies were found, however, close to those positions in the flow where the flow velocity changed sign and in the separated region of the boundary layer.

Evolution of disturbances in three-dimensional boundary layers

An integral calculation method for incompressible three-dimensional boundary layers in axisymmetric diffusers

Steady incompressible magnetohydrodynamic mixed convection boundary layer flow of a Casson fluid on an exponentially vertical shrinking sheet using the non‐Newtonian heating equation is investigated in this paper. There are three main objectives of this study, namely, to develop a new mathematical model, to obtain multiple solutions, and to perform stability analysis. The governing partial differential equations have been changed into nonlinear ordinary differential equations. The resultant equations of boundary value problems are then converted into the equivalent initial value problems using the shooting method before they can be solved using Runge‐Kutta of order four. The numerical results are obtained and found to be in good agreement with the published literature. The results also indicate that the velocity boundary layer becomes thinner as the magnetic, slip, and Casson parameters increase. Dual solutions for temperature and velocity distributions are obtained. Furthermore, the results suggest that the presence of the force of buoyancy (opposing flow case) would cause the occurrence of dual solutions. However, based on the stability analysis, only the first solution is stable.

Experimental results are presented that reveal key features of the large-scale organized structures in a supersonic, turbulent boundary layer. Measurements were obtained in a Mach 3 zero-pressure-gradient boundary layer using a crossed-wire probe and arrays of normal hot wires with vertical, spanwise, and streamwise separations ranging from 0.1 to 0.6δ. Space–time correlation results indicate the existence of large-scale structures of a size comparable to δ, with a spanwise extent only slightly less than the vertical scale. The convection velocity of the large-scale motions is nearly constant across 80% of the boundary layer and is equal to approximately 0.9U∞. It is shown that positive events detected with the VITA conditional sampling technique correspond to steep gradients in the streamwise mass flux which extend across most of the boundary layer. These sharp gradients appear to be the upstream interfaces of large-scale turbulent ‘bulges’, similar to those seen in incompressible boundary layers. In a reference frame moving with the convection speed of the sharp gradients, low-momentum fluid is observed rotating on a large-scale, while high-momentum fluid forms a saddle point on the upstream edge of the large-scale motion. These motions are associated with elevated levels of the shear product, emphasizing their role in the dynamics of the boundary layer.

Hot-wire anemometry measurements in an incompressible turbulent boundary-layer flow over a flat plate, at zero pressure gradient were made using two X-probes simultaneously. The over seven meters long flat plate installed within the wind tunnel generated an approximately 20-cm thick boundary layer, and R(theta) of about 4,000. The mean velocity and turbulence intensity data very close to the wall were measured with a single-wire. Space-time velocity fluctuation and shear-product correlation measurements enabled the calculation of large-scale structure convection velocity, inclination-angle to the wall, and spatial extent in space. The shear-product correlation results are reported for the first time.

Experiments described here show that the rate of heat transfer in a supersonic, turbulent, boundary layer on a concave wall is increased by the streamwise curvature of the wall. For the arrangement investigated, the pressure is kept constant along the wall, and the increase of about 20% is therefore only due to the wall curvature. For a convex wall, a comparable decrease was found, also with constant pressure along the wall. It is likely that this change in heat-transfer rate is mainly due to an increase or decrease of turbulent mixing by the effect of the curvature. The increase on a concave wall can in principle also be explained by large-scale vortices with axes in the flow direction (Görtler vortices). However, disturbances of this type cannot explain the decrease observed on the convex wall. They cannot therefore be the only cause. A simple criterion indicating the effect of curvature on the turbulent motion normal to the wall is given. It is derived from an inviscid-flow analysis. The criterion shows that, for most Mach numbers and wall temperatures of practical interest, the change in mixing depends mainly on the velocity gradient normal to the wall. For high supersonic and hypersonic Mach numbers, however, there is a layer near the outer edge of the boundary layer where the change depends mainly on the temperature (density) gradient.

Direct numerical simulation of incompressible turbulent boundary layers and planar jets at high Reynolds numbers initialized with implicit large eddy simulation