Stokes Layers in Horizontal- Wave Outer Flows J. E. Choi Graduate Research Assistant, Results are reported of a computational study investigating the responses of flat plate Presently, Research Scientist, boundary layers and wakes to horizontal wave outer flows. Solutions are obtained Hyundai Corporation. for temporal, spatial, and traveling waves using Navier Stokes, boundary layer, and perturbation expansion equations. A wide range of parameters are considered for all the three waves. The results are presented in terms of Stakes-layer overshoots, M. K. Sreedhar phase leads (lags), and streaming. The response to the temporal wave showed all Postdoctorai Associate. the previously reported features. The magnitude and nature of the response are small and simple such that it is essentially a small disturbance on the steady solution. Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/118/3/537/5900332/537_1.pdf by University Of Iowa user on 26 February 2021 Results are explainable in terms of one parameter ^ (the frequency of oscillation). F. Stern For the spatial wave, the magnitude and the nature of the response are significantly Professor and Research Engineer. increased and complex such that it cannot be considered simply a small disturbance on the without-wave solution. The results are explainable in terms of the two parame­ Department of IVIechanical Engineering/ ters \~' and x/k (where \ is the wavelength). A clear asymmetry is observed in the Iowa Institute of Hydraulic Research wake response for the spatial wave. An examination of components of the perturbation The University of Iowa, expansion equations indicates that the asymmetry is a first-order effect due to nonlin­ iowa City, Iowa 52242-1585 ear interaction between the steady and first-harmonic velocity components. For the traveling wave, the responses are more complex and an additional parameter, c (the wave speed), is required to explain the results. In general, for small wave speeds the results are similar to a spatial wave, whereas for higher wave speeds the response approaches the temporal wave response. The boundary layer and perturbation expan­ sion solutions compares well with the Navier Stokes solution in their range of validity. Introduction show significant wave-induced effects such as three-dimen­ sional boundary-layer and wake responses with large overshoots In a variety of applications, boundary layers and wakes are and phase shifts, wave-induced separation, asymmetric wake subject to oscillating outer flows. Of interest here are outer responses with increased overshoots for regions of favorable flows of uniform streams with superposed temporal, spatial, pressure gradient and anisotropic/nonequilibrium nature of tur­ or traveling waves. Example applications are surface-piercing bulence. bodies and turbomachinery. A basic geometry is a flat plate As in the case of spatial waves, very limited information is aligned with the uniform stream. available for traveling horizontal-wave outer flows except for The most extensive information is for laminar and turbulent the work of Patel (1975, 1977). Approximate PE computations boundary layers with temporal horizontal-wave outer flows and data for laminar and turbulent boundary layers show in­ (Telionis, 1981). Boundary-layer (BL) and perturbation- creased overshoots and phase lags in comparison to the temporal expansion (PE) equations computations and data show that the wave. The above studies also indicate a dominant effect of the flow is characterized by small-amplitude Stokes-layer over­ wave speeds on the response for both laminar and turbulent shoots, phase leads, and streaming. Except for high-frequency boundary layers. However, the measurements were obtained and/or strong-separation conditions, limited interactions occur only for a single wave speed and low and medium reduced between the mean and turbulent motions, i.e., the turbulence frequencies which somewhat limits the scope of the conclusions exhibits a quasi-steady response. Some of the recent numerical made. Only very limited turbulence measurements were made investigations on temporally oscillating turbulent boundary and they were found to be independent of the reduced frequen­ layer flows are a direct numerical simulation by Spalart and cies considered. Baldwin (1989) and a Reynolds-averaged Navier Stokes (RANS) calculation by Hanjalic et al. (1993). These new stud­ Also of interest is related work for traveling vertical- or com­ ies also confirm the quasi-steady response of temporally oscil­ bined-wave outer flows; however, in distinction from the previ­ lating transitional-turbulent boundary layers with a marked tran­ ous cases, unsteady lift effects predominate. Classic inviscid sition phase followed by a laminarization phase in each cycle. analytic solutions predict the unsteady lift amplitude reduction and phase shifts (Horlock, 1968). Data for foils with combined Relatively limited information is available for spatial waves waves have confirmed the theory for low frequencies (Poling and has yet to be put in the context of Stokes layers. Stem and Telionis, 1986). Recent experiments (Lurie, 1993) and (1986), Stern et al. (1989, 1993), and Choi and Stem (1993) computations (Paterson and Stem, 1993) concern the complex investigated the boundary-layer and wake of a surface-piercing nature of the boundary-layer and wake response, especially near flat plate with a Stokes-wave outer flow which included detailed the trailing edge and in the near wake. towing-tank experimental measurements. The plate and wave There appears a need to investigate, compile, and present the lengths are equal and the wave phase is fixed such that crests effects of different outer wave flows on boundary layers and are coincident with the leading and trailing edges. Computation wakes using computational and experimental techniques so as of BL, Navier-Stokes (NS), and RANS equations and data to present a unified picture. The present work attempts to do the computational part for laminar flows which is clearly an initial step taking into account the need to consider turbulent Contributed by the Fluids Engineering Division for publication in the JOURNAL boundary layers and wakes. Thus, the primary objective of the OF FLUIDS ENGINEERING . Manuscript received by the Fluids Engineering Division February 17, 1994; revised manuscript received April 23, 1996. Associate Techni­ present work is to determine the similarities and differences cal Editor; S. P. Vanka. between the laminar boundary layer and wake responses due to Journal of Fluids Engineering SEPTEMBER 1996, Vol. 118/537 Copyright © 1996 by ASME U, — , p\ = (Ue, - — , Pe\ .05 < X < Xexil, ^ = S(x) d^u dp \J tAr »^e> dx^ ' dx s, - (Exit Plane) (M, V) = (UB, VB) X = .05, f = 0 (4) where (MB, Dg) is the Blasius solution. The symmetry boundary condition on the wake centerline imposes a strong restraint on the wake development and is suited only for laminar flows as Fig. 1 Domain for boundary-layer wake computations viiVn temporal-, is the case in the present work. Also, the impostion of the spatial-, and traveling-wave external flows. For boundary-layer calcula­ Blasius profiles at the inlet is justified for the Reynolds numbers tions the plate extends to the exit plane. used in this study. Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/118/3/537/5900332/537_1.pdf by University Of Iowa user on 26 February 2021 U, = I + Ui cos [^(t - x/c)] (5a) temporal, spatial, and traveling horizontal-wave outer flows. such that Another objective is to identify the physical mechanism respon­ sible for the previously mentioned asymmetric wake response. t/.e To these ends, a model problem of a flat plate boundary layer V, = y sin [^(t - x/c)] (Sb) and wake was selected and NS, BL, and PE solutions are ob­ tained for a large range of outer flow conditions. -dpjdx = ^t/i( 1 - - I cos [^{t - x/c) + 7r/2] Model Problem Consider the model problem of a flat plate laminar boundary + MU} sin [2^(t - x/c)] (5c) layer and wake in an outer flow of a uniform stream with superposed temporal, spatial, or traveling waves (Fig. 1). The continuity and NS equations for unsteady two-dimensional in­ -dpjdy= -ey,( 1 -- compressible flow are written in Cartesian coordinates with x positive downstream, y normal to the plate, and (0, 0) at the plate leading edge: X^cos[iit - x/c)] + (^] y {5d) du (1) ox. +1ay^ = 0 Pe = Ui(C - 1) cos [^(t - x/c)] du du du dp J_ d^u d^u + U 1" t^ TT (2) - .25Ujcos[2^{t - x/c)] (5e) dx Re dy^ dt dx dy dv dv dv 1 { d\ uj, = { - \ Uiy cos [C(f - x/c)] (5/) h M hi) dt dx dy dy Re a?-^V' ^'^ Ui is the unsteady velocity amplitude, f = UJL/UO is the fre­ where {u, v) are the velocity components in the {x,y) direc­ quency parameter (to is the frequency), and c = k/T = ^/A; is tions, p is the pressure, and Re = UoLlv is the Reynolds num­ the wave speed [ \ is the wave length, T is the wave period (= ber. All variables are nondimensionalized using Uo, L, and p. 2n/Q and Ic is the wave number (= 27r/X)]. Equation (5) The boundary and initial conditions are: corresponds to the traveling wave. For the specified [/^ in (5a), relations (5b)-(5d) are obtained from (l)-(3), respectively, dp M, D, = 0 .05 s X s 1, y = 0 (5e) by integration of (5c), and (5/) by definition uie = — dWJ d^ dx using (5i>). The temporal wave is recovered for c -* oo, i.e., A; = 0: du dp — , V, -^ 0 1 < X < x„ y = 0 t/^ = 1 -I- f/i cos ^f (6fl) dy dy Nomenclature c = wave velocity Uo = uniform-stream velocity ^ = frequency k — wave number Ui = (I) unsteady velocity amplitude ui = dimensional frequency L = reference length = (2) perturbation parameter 0( ) = order of magnitude ', y = Cartesian coordinates Subscripts p — pressure a = phase-shift angle cl =wake centerline Re = Reynolds number (= UoL/v) 6 = boundary-layer or wake thickness e = edge value t = time 5^ = Stokes-layer thickness m = maximum value T = wave period •y = phase angle 0 = steady solution u, V = boundary layer or wake veloc­ X = wave length s = streaming ity u = kinematic viscosity 1 = first harmonic U, V = external-flow velocity 538 / Vol.