The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

An introduction to heat transfer


 The spectral ratio between horizontal and vertical components (H/V ratio) of microtremors measured at the ground surface has been used to estimate fundamental periods and amplification factors of a site, although this technique lacks theoretical background. The aim of this article is to formulate the H/V technique in terms of the characteristics of Rayleigh and Love waves, and to contribute to improve the technique. The improvement includes use of not only peaks but also troughs in the H/V ratio for reliable estimation of the period and use of a newly proposed smoothing function for better estimation of the amplification factor. The formulation leads to a simple formula for the amplification factor expressed with the H/V ratio. With microtremor data measured at 546 junior high schools in 23 wards of Tokyo, the improved technique is applied to mapping site periods and amplification factors in the area.

Ground Motion Characteristics Estimated from Spectral Ratio between Horizontal and Verticcl Components of Mietremors.

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

A direct numerical simulation of incompressible channel flow at a friction Reynolds number ( ) of 5186 has been performed, and the flow exhibits a number of the characteristics of high-Reynolds-number wall-bounded turbulent flows. For example, a region where the mean velocity has a logarithmic variation is observed, with von Karman constant . There is also a logarithmic dependence of the variance of the spanwise velocity component, though not the streamwise component. A distinct separation of scales exists between the large outer-layer structures and small inner-layer structures. At intermediate distances from the wall, the one-dimensional spectrum of the streamwise velocity fluctuation in both the streamwise and spanwise directions exhibits dependence over a short range in wavenumber . Further, consistent with previous experimental observations, when these spectra are multiplied by (premultiplied spectra), they have a bimodal structure with local peaks located at wavenumbers on either side of the range.

Direct numerical simulation of turbulent channel flow up to

Purpose




This paper aims to show results of numerical simulations of transonic flow around a supercritical airfoil at chord Reynolds number Rec = 3 × 106, with the aim of elucidating the mechanisms responsible for large-scale shock oscillations, namely, transonic buffet.




Design/methodology/approach




Unsteady Reynolds-averaged Navier–Stokes simulations and detached-eddy simulations provide a preliminary buffet map, while a high fidelity implicit large-eddy simulation with an upstream laminar boundary layer is used to ascertain the physical feasibility of the various buffet mechanisms. Numerical experiments with unsteady RANS highlight the role of waves travelling on pressure side in the buffet mechanism. Estimates of the propagation velocities of coherent disturbances and of acoustic waves are obtained, to check the validity of popular mechanisms based on acoustic feedback from the trailing edge.




Findings




Unsteady RANS numerical experiments demonstrate that the pressure side of the airfoil plays a marginal role in the buffet mechanism. Implicit LES data show that the only plausible self-sustaining mechanism involves waves scattered from the trailing edge and penetrating the sonic region from above the suction side shock. An interesting side result of this study is that buffet appears to be more intense in the case that the boundary layer state upstream of the shock is turbulent, rather than laminar.




Originality/value




The results of the study will be of interest to any researcher involved with transonic buffet.

Scrutiny of buffet mechanisms in transonic flow

We study the fluid dynamics of rolling wheels at Reynolds number (where is the Reynolds number based on the wheel diameter), with the objective of characterizing the various regimes of steady and unsteady motion. Regardless of the Reynolds number, the flow is found to separate approximately upstream of the apex of the wheel, where a saddle point in the pseudo-streamtrace pattern is observed. Under the flow conditions here essayed, the drag coefficient steadily decreases with , and the lift coefficient remains strictly positive. The positive lift provided by the rolling wheel is associated with the presence of a strong (positive) peak of the static pressure in the upstream proximity of the contact point with the ground, which we interpret as the result of the impingement of flow particles entrained in the boundary layer that develops on the front part of the wheel. Steady laminar flow is observed up to , which is characterized by a three-dimensional wake whose length increases with the Reynolds number. Unsteadiness is first observed at , under which conditions the flow retains planar symmetry, and is characterized by the quasi-periodic shedding of hairpin vortices. Transition to three-dimensional flow happens at , in which case a sinuous mode of instability in the wheel wake is established, which modulates the shedding of the hairpins, and which causes the onset of a non-zero side force. At the highest Reynolds number considered here () the wake exhibits some characters of turbulence, with wide-band frequency spectra, and its topology entirely changes, becoming split into two parts, and being much shortened compared to the lower- cases. Despite the limitation of the study to low Reynolds numbers we find that, once significant three-dimensionality and scale separation are established in the wheel wake, the nature of the flow becomes qualitatively similar to fully developed turbulent flow. In perspective, this observation opens interesting avenues for the prediction of unsteady flow around rotating tyres at moderate computational cost.

The fluid dynamics of rolling wheels at low Reynolds number

In the present paper the dynamics of the coupling process of compressible vortex pairs is analyzed by means of extensive numerical simulations. The objective of the study is to determine the effects of the initial vortex spatial structure and of the compressibility. Different vortex structures have been considered and their influence on the vorticity dynamics has been assessed. In the case of free evolution, the numerical simulations show that two vortices in close contact evolve toward a vortex dipole; however, the structure of the resulting dipole depends upon the initial inner core structure and on the vortex intensity. In the presence of shock wave–vortex pair interaction the coupling process depends upon the strength of the shock and the type of pair. In the case of a colliding pair, the process obeys either a two- or a three-stage mechanism depending upon the shock strength, and nonlinear couples form regardless of the initial vortex Mach number. If the interacting shock is strong, the vortex pair e...

Simulations and analysis of the coupling process of compressible vortex pairs: Free evolution and shock induced coupling

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU
 w
 ≡ 2U
 w
 , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity u
 τ
 , and h is the half-height of the channel. This leads to a one-parameter family of one-dimensional flows of varying Reynolds number Re ≡ U
 w
 h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of two each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend’s hypothesis of universal behavior for the velocity and shear stress, when they are normalized with u
 τ
 and h; on the other hand universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford’s conjectures, suggesting that at least in this idealized flow turbulence theory is successful like it was for the classical logarithmic law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, that again does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.

/pdf/direct-numerical-simulation-and-theory-of-a-wall-bounded-11a1tupul9.pdf

Sergio Pirozzoli

Papers

Scrutiny of buffet mechanisms in transonic flow

The fluid dynamics of rolling wheels at low Reynolds number

Simulations and analysis of the coupling process of compressible vortex pairs: Free evolution and shock induced coupling

Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction

Natural grid stretching for DNS of wall-bounded flows