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. 118, SEPTEMBER 1996 Transactions of the ASME such that Computational Methods -dpe/dx = ^U,cos(0 + n/2) i6b) The NS solutions are obtained using the methods of Choi and Stern (1993) and Paterson and Stern (1993) for steady and Pe= - £,UxX cos (C? + 7r/2) (6c) unsteady flows, respectively. Both methods are based on Chen and Patel (1989, 1990), which have been extensively validated which implies a phase lead retarded by u,. For the temporal for a wide range of applications. The NS and continuity equa wave {5b), (5d), and (5/) are identically zero. tions are written in the physical domain using Cartesian coordi The spatial wave is recovered for c -» 0, i.e., ^ = 0: nates (x, y, z) and partially transformed (i.e., coordinates only) into nonorthogonal curvilinear coordinates (^, rj, C,) such that Uc = I + Ui cos (a — kx) (la) the computational domain forms a simple rectangular parallel epiped with equal grid spacing. The transformed equations are such that reduced to algebraic form using finite-analytic method spatial and second-order backward finite-difference temporal discreti Ve = -kUiy sin (a - kx) (lb) zation. SIMPLER and PISO velocity-pressure coupling algo 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 -dpjdx = -kUy cos [a - kx -\- ir/l] rithms are used for steady and unsteady flow, respectively. The resulting implicit equations are solved using line-ADI with un- + .SkU\ .sin [2(a - kx)] (1 c) der-relaxation. The boundary and initial conditions are given in (4). The details of the edge values for different wave conditions -dpeldy = k'-U.y cos (a - fc;<;) + euly (Id) are given in (5)-(7). The BL and PE equations are derived following Telionis p, = -Ui cos (a - kx) — .ISU] cos [2(a — kx)] (le) (1981). The derivation is complicated and, as already noted, uje = k^U,y cos (a - kx) (If) requires (i~\ X~', ^/Re, \/Re"^) < 1. In consideration of the relative simplicity of the resulting equations, a straightforward which implies a phase lag enhanced by MM, . Here a is the phase- finite difference method is used with l^rst-order backward differ shift angle. ences for the t- and x-derivatives, a central difference for the The solutions are restricted to large Re and small C/i and y-derivatives, and a lag-iterative algorithm for the nonlinear ^Ui(l - 1/c) such that viscous-inviscid interaction is negligi terms. The PE equations are solved successively and further ble and, additionally, for (C', X-"', •?/Re, X/Re"') -^ 1, the analyzed through presentation of average values over the bound BL and PE approximations are valid. The formulation can be ary-layer and wake thickness of the terms in the first-order extended to finite-thickness geometries (e.g., foils) such that equation U„ = Uo{x, y); however, incorporation of viscous-inviscid in teraction is required and the applicability of the BL and PE lii - kUo)ui + M„ —- + Ml -—- + i;„ -—• + Ui -— solutions is more restricted. ox ax ay oy The results are analyzed by decomposing u(x, y, t[a]) into steady (i.e., with-out wave) Uo(x, y), streaming Us(x, y), and = i(^-k)U,+^^ (12) first-harmonic response amplitude u,(x, y) and phase y(x, y) Re oy components [where (u,, u,) are real for the zeroth-order (i = 0) and complex M(X, y, t[a]) = u„(x, y) + u,,(x, y) for first-order (; = 1) ]. The averaged value is obtained as + U\(x, y) cos [(,t[a] - kx + y(x, y)] (8) (l>i(x) = -( Wdy (13) 0 Jo where where | <^i | (1 = 1, 9) is the magnitude of the ith term in (12) J /•r|2,rl with the index increasing from left to right. The application of Mx, y) = TT—— u(x, y, t[a])dt[da] - u„(x, y) (9) boundary conditions and initial conditions are straight forward TIZTT] JO and details are given in Choi (1993). u,(x, y) = \ ——• I \u(x, y, t\a\) Computational Conditions and Uncertainty Analysis Computations were performed to cover a wide range of pa rameters for temporal, spatial and traveling wave outer flows. — u,,(x, y) — u„(x, y)]^dt[da] (10) The different computational conditions are summarized in Table 1. The range of values were specified both for validation through comparisons with previous studies and for investigation of the _2 flTllTT] influences of the wave and pressure-gradient amplitudes and sin y(x, y) [u(x, y, t[a]) wave frequency, length and speed. For the temporal wave, for T[2n]i]ui(x, y) Jo the boundary-layer region, NS, BL, and PE solutions were ob - u,(x, y) - u„{x, y)] sin {(,t[a\ - kx)dt\da] (Ua) tained for a large range of frequencies (.01 s ^ < 5). For the wake region, only BL and PE solutions were obtained for 2 PT[2i,] relatively low frequencies (.19 =s ^ < .94). For the spatial cos y{x, y) = „„ i , 7 [u(x,y,t[a]) wave, for both the boundary-layer and wake regions, NS, BL, T[2ir\Ui(x, y) Jo and PE solutions were obtained for a range of wavelengths (.2 — Us{x,y) - M„(x, y)] cos (£,t[a] — kx)dtYda\ (\\b) < X."' s 10). For the traveling wave, for the boundary-layer region, solutions were obtained for large ranges of frequency and [a], \da\, and [27r] replace t or (,t,dt, and T, respectively, and wave speeds (.1 < if < 5 and .3 s c < 10) with NS for the spatial wave. Figures are presented for M|/(/J and y vs. solutions limited to low frequencies. For the wake region, only rj (= y/VRe^, where Re;t = U„xlv). Also presented are the BL and PE solutions for relatively low frequencies (.19 < ^ < maximum overshoot values UiJU\, it's position 7ji„, and wall .94) and for wave speeds (.3 s c < .10) were obtained. For or wake-centerUne phase angles y^ vs. ^ = ujxl [/„ for the tempo fixed wavelengths (\'' = .2 and 1) solutions were obtained ral and traveling waves and vs. xl\ for the spatial waves. with all the three methods for a wide range of frequencies
Journal of Fluids Engineering SEPTEMBER 1996, Vol. 118/539 Table 1 Calculation conditions. BL and PE solutions obtained for all the above values. NS solutions obtained only for the underlined values Temporal wave Region Wave/pressure gradient amplitude Frequency (Q Boundary layer (1) U, = -0.4 and (2) ^C/, = -0.13 .01, .05, i 2, .3, .4, .522, ,559, .6, .7, .8, .9, 1.0 1.1, 1.2, 1.3, 1.4, L47, 1.6, 1.7, 1.8, 1.9, 2.0, 2.2, 2.49 2.65, 2.8, 3, 3.2, 3.4, 3.6, 3.8, 4, 4.2, 4.4, 4.6, 4.8, 5.0 Wake (3) £/, = -0.4 and (4) ^C/, = -0.13 .188, .565, .942
Spatial wave Region Wave/pressure gradient amplitude Wavelength^' (\"') Boundary layer and wake (1) t/, = -0.4 and (2) kUi = -0.13 ^ U 5^10, 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 Traveling wave
Region Wave/pressure gradient amplitude Frequency (JVwave speed (c) Boundary layer (1) [/i = -0.4 ? = .314, .628, .943. 1.257, 1.57 c = .3, .4, .5, .64, .77, 1., 2., 5., 10. Boundary layer (2) I/i = -0.4 5 = .1, .2, .522, .599, 1., 1.47, 2.49, 2.65, 5. c = .3, .5, .77, 1., 10. Wake (3) Ut = -0.4 $ = .188, .565, .942 c = .64, .77, 1., 10. Boundary layer and wake (4) \-' = 0.2 (c,0 = (A :251), (^ 377), (A ,503), (/77, .968), ec/,(l - 1/c) = -0.013 (2^ 2.513), (5^ 6^) Boundary layer and wake (5) \-' = 1.0 (c,0 = (A 1-257), (3, 1.885), (4, 2.513), (.77, 4.838), et/,(l - 1/c) = -0.013 (2, 12.566), (5j 31.416) for both boundary layer and wake. For the above parameters, order finite-difference methods. A numerical study using gener solutions were obtained keeping either the unsteady velocity alized Richardson extrapolation for a laminar flat-plate and fully amplitude constant or the amplitude of the pressure gradient developed annular flow seem to support this conclusion. Also constant. In summary, a large range of conditions were evalu note that, a few of the cases reported in this paper were repeated ated for all three waves and methods; however, the ranges for recently using a third-order finite difference scheme and similar the wake region and NS solutions are relatively restricted being results were obtained. more computationally intensive. Due to limitations of the range The grids were generated algebraically. A range of grids and of validity of the different approximations, especially for the computational domains were used depending on the method, BL and PE equations, each of the three techniques are often region of interest (i.e., boundary layer or wake) and \"'. The used to complement each other in presenting a unified picture. solution domain for the wake calculation is indicated in Fig. 1. However, every attempt is made to compare the results obtained For the boundary layer calculations the plate extends all the using the three techniques wherever the validity of the approxi way to the exit plane. The grid numbers, exit plane and time mations permits. Although limited by approximations, the BL steps were determined based on grid independence and time- and PE approaches are also essential to the present study mainly accurate studies for selected conditions (Choi, 1993). The because of the following reasons. First, they are inexpensive shortest wave considered (\ ' = 10) was described by at least which enables us to obtain solutions for a wide range of parame 20 points in the axial direction. 100 time steps per period were ters within their range of validity. Second, certain aspects of used. The details are summarized in Table 2. The leading edge the flow are better understood by these methods, such as the PE was not included in the domain while trailing edge resolution method which allows us to understand the interactions between is given in Table 2. The convergence criterion was based on various components in the equations. the residual The solutions are for Re = 10', except the temporal and traveling wave NS solutions are for Re = 10". Numerical tests R(it) = 11 (\(it - 1)1 - \
540 / Vol. 118, SEPTEMBER 1996 Transactions of the ASME Table 2 Grid information Boundary layer and perturbation expansion equations
Steady BL Steady BL Unsteady BL Steady BL Steady BL and wake and wake Unsteady BL and wake Parameters l/\ = 0.2, 1 l/\ = 5, 10 l/\ = 0.2, 1 l/\ = 5, 10 alU aiU Inlet bound.(j:/L) 0.05 0.05 0.05 0.05 0.06 0.06 Outlet bound.(x/Z,) 1.50 1.50 10.00 5.00 1.50 5.00 AJC/L(T.E.) 0.001 0.005 0.01 0.005 0.02 0.02 Top bound.ty/6) 1.26 1.26 1.26 1.26 1.26 1.26 Ay/fi (even grid) 0.0124 0.0124 0.0124 0.0124 0.06 0.0155 Grid no. (X-dir) 146 291 996 991 76 248 Grid no. (y-dir) 101 101 101 101 51 81
Navier stokes equation 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 Steady BL and wake Steady BL and wake Unsteady BL and wake Parameters l/\ = 0.2, 1 1/X = 5, 10 all? Inlet bound.(;c/Z,) 0.05 0.05 0.05 Outlet bound.(x/L) 10.00 5.00 5.00 Ax/Z-(T.E.) 0.016 0.01 0.009 Top ho\xnd.(y/S) 1.26 1.26 1.26 Ay/6 (min) 0.015 0.01 0.00004 Grid no. (X-dir) 171 495 150 Grid no. (K-dir) 23 23 75 starting point, which is defined as the initial axial position for to describe the results. The results are identical for the same which reverse flow occurs. values of £ whether £ = uiL/Uo and varying ui, or ^ = LJX/U„ and varying jc < L. To see the effect of £ on separation, separa Temporal Wave tion starting points were obtained for the different £ values. It was found that separation is delayed for increasing £. Most of Boundary-Layer Response. The response of the boundary the above features of the flow were discussed previously by layer to a temporal wave displayed all the well known features Telionis (1981), including, additionally, that increasing con of the flow: velocity overshoots, phase leads, and streaming. stant steady adverse pressure gradient increases the velocity While the velocity overshoots and phase leads showed good overshoot amplitudes. comparison with the data, streaming was almost zero and no The differences between the three solutions for MI , y, and the data was available. The magnitude and nature of the response separation starting points are relatively small. The NS solution were small and simple such that the response can be considered showed larger and smaller maximum velocity overshoots for as essentially a small disturbance on the steady solution. Shown small and medium f, respectively (i.e., a less pronounced peak in Fig. 2 are u,„/Uu Vi«>, and -y„ vs. £ (= tJx/U„), including spread over a larger region). The differences between the solu comparisons with the data (Hill and Stenning, 1960) and previ tions for Mj (not shown here) were significant with the PE ous calculations (Telionis, 1981). The position of the overshoot solution displaying complex profiles with large £ dependency. monotonically decreases from low to high ^. At the wall and Note that the differences between the NS and BL and PE solu in the inner part of the boundary layer, the phase angle increases tions for small £ are expected; since, the order-of-magnitude linearly for small £ and then merges with the high £ value of estimates used in deriving the latter two equations require 45 degrees. It was found that the single parameter £, is sufficient 6J6 = 0(1), which is violated (i.e., for small £, 6J6 > 1). Wake Response. Some typical results of the response of the wake to a temporal outer wave are displayed in Fig. 3 O Tellonls A Hill and Stenning which shows Ui/Ut and y versus r] profiles for three values of NS, Re=10'' BL, Re = 10^
—I— 4.0 60- 1 5 =0,168 2 i =0.565 30- 3 5 =0.942 / 2
i'
oU -\ 1 1 1 1
Fig. 2 Maximum first-harmonic overshoot values and wall phase angle Fig. 3 First-harmonic amplitude and phase atx = 1.5: wake, temporal values: boundary layer, temporal wave wave (BL equation solutions shown)
Journal of Fluids Engineering SEPTEMBER 1996, Vol. 118/541 0 2.0 4.0 6,0 45 -, ., \ = 0.2 \ =1.0
0 - -,^=^
45 -
90 - 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 2.0 4.0 6.0 () 2.0 4.0 6.0
Fig. 5 First-harmonic amplitude and phase atx = 1.0: boundary layer, spatial wave
the response is relatively small, whereas for medium \."' = 1 (and 5) the response is extreme. Some typical curves are shown 2.0-1 (c) t = 0.565 in Fig. 5 which displays Ui/Ui and y vs. rj profiles for the wave 1 Itu, 4 u„,u, 7 itU, length parameters k~' =0.2 and 1 at x = 1. In both cases, the m 1.5- 2 ikUpU, 5 VgU^y 8 ikU, overshoots are considerably larger than the temporal wave.
§1.0- Large phase lags are observed, as are streaming velocities (not >-9 E shown here). In the outer part of the boundary layer, the phase o O 0.5- is nearly 0. For \~' = .2 and 1, the NS solution phase-angle 9,6,9,4 and 3 ^6 response is similar to that for the BL and PE solutions. However, ^ 1 1 _|, ^/1 , 1 ,„ 1 1 _! 1 for higher values of \ ' (not shown here) there are significant 4.0 6.0 8.0 deviations. X Shown in Fig. 6 are UiJUx, r^i™, and y^ versus xl\ for two Fig. 4 Wake response: temporal wave, (a) wake centerline velocity, (b) values of X"' (0.2 and 1). Also shown in Fig. 6 is velocity defect, (c) averaged values of perturbation-expansion equation components for ^ = .565. (BL equation solutions sliown in a and b.) Mi„/(f/ivRAc) versus xl\ for the same two values. The maxi mum overshoot values are seen to require l/vRe^ scaling, be cause the values monotonically increase with xl\ unless nor frequency parameter ^ = (.188, .565, .942)atjt= 1.5, including malized by a6 (where a is a constant) whereas Mi„/(f/iVRej:) comparison with the steady solution. The BL and PE solutions displays a maximum overshoot near;c/\ = 1 and then monotoni are similar (except for the streaming profiles, not shown here, cally decreases toward its asymptotic value. For X^' = 5 and which are negligible) and display trends as described earlier 10 (not shown here) the trends are similar both with and without for the boundary-layer, but with reduced response. Here again, scaling, i.e., both UiJUi and uiJiUiS^t^) display maximum the nature of the response is essentially a small disturbance on overshoots near x/X = 1 and then monotonically decrease to the steady solution. The profiles of Ui„/Uu Usm, Vim, V^m and wards their asymptotic values, which decrease with increasing 7o vs. ^ (not shown here) all displayed similar trends in Fig. 2 X"'. Recall that l/VRe^ scaling is required for v„{x, y). over the plate, but with some trailing edge effects. In the far In addition to the differences between the solutions already wake, the velocity overshoots, streaming and phase leads mono- described, the NS solutions are found to display smaller over tonically decrease such that ultimately a plug flow is achieved. shoots. These differences between the NS solutions and BL and Figures 4(a) and (b) show the wake-centerline velocity Auci/Ui and the defect 1 — \Auci/Ui\ (where Au^ = u^ — Uoci - u.fci)- Consistent with Ui/Ui, the response is similar to the steady solution, but with faster recovery for increasing f. For the larger ^ values, recovery is achieved at .* - 1 = 2.75. The wake response is symmetric. Shown in Fig. 4(c) is the boundary layer and wake averaged values of the first-order PE Eq. (12) for ^ = .565. In the far wake, the terms 1 and 7 are the dominating terms and are in balance (i.e., MJ = Ui).
Spatial Wave Boundary-Layer Response. As was the case for the tem poral wave, the spatial wave is also seen to be characterized by velocity overshoots, phase lags (in this case), and streaming; however, there are significant differences between the two outer flows. The magnitude and the nature of the response are signifi cantly increased and complex such that it cannot be considered simply a small disturbance on the without-wave solution. The dependency of the response on X~' was found to be very strong, i.e., the characteristics of the response vary significantly for the Fig. 6 lUaxImum first harmonic and first harmonic wall-phase angle specified values of X"'. For small and large X~' (.2 and 10), values: boundary layer, spatial wave
542 / Vol. 118, SEPTEMBER 1996 Transactions of the ASME -a =0° - a = 180° 5w^^0m^0mm:k.
(0) —I 4.0 6.0 8.0 10.0 [x-1]/\
4-1 LA =1.0 \''=1.0
1 " '^NJ^ maximum t diHerence 1—1—\—I—I—I—I ?°- •0 minimum 0 2.0 4.0 6.0
s - 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 Fig. 7 First-harmonic ampiitude and phase at x = 1.5: wake, spatial , (b) 1 1 1 1 wave 10.0
PE solutions are seen to increase for increasing X~'. This is clearly evident from Figs. 5 and 6. Note that deviations between the NS solution and BL and PE solutions for large \~' are expected; since, the order-of-magnitude estimates used in deriv ing the latter are no longer valid. Unlike the temporal wave, a single parameter is not sufficient to characterize the results, i.e.,
both x/\ and X."' are required as is l/VRe;t scaling. The separa I I , tion starting point was computed for each value of \~' and the 8.0 10.0 results indicated that separation is delayed for increasing X.^'. Fig. 8 Wal
Journal of Fluids Engineering SEPTEMBER 1996, Vol. 118/543 data and previous solutions indicates the need for additional investigations; both computational and experimental. Figure lO(fl) shows Mj/t/j and y versus i] for fixed c and variable ^. The c = .3 trends are similar and consistent with those for the spatial wave for the range of ^ (i.e., \"') investi gated, which, unfortunately is restricted to long and medium wave lengths. It was not possible to obtain BL and PE solutions for small wavelengths due to the required large (JJ\{\ — (1/ c)) and associated large response amplitudes. The c = 10 trends are similar and consistent with those for the temporal wave, in this case, for the complete ^ range of interest. For large ^ and c = 1 (not shown here), Mi/f/i displays a nearly linear variation across the boundary layer and y displays large leads. For small ^, the profile is somewhat fuller, although there is no overshoot,
and y is nearly zero. The streaming velocity (not shown here) 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 is minimum. The intermediate c values indicate a smooth transi Fig. 11 First-harmonic amplitude and phase at x = 1.5: wake, traveling wave (BL equation solutions shown) tion between the spatial and temporal waves. In Fig. lO(fc) is shown uxllJ\ and y versus r\ for fixed ^ and variable c. The small ^ response indicates negligible influence of c (i.e., \"') for the long-wave range investigated compared to large i^ re Wake Response. The wake response also showed the lim sponse. iting responses for small and large wave speeds, with the tempo ral wave response recovered for the large wave speeds and the Profiles of maximum overshoot values and their positions spatial wave response for smaller wave speeds. Note that for and wall phase angle values (not shown here) also displayed the specified f range, c — .64 appears to be a mixed condition, similarity to the spatial and temporal waves for low and high i.e., in the transition range between the spatial and temporal values of c, respectively. So also did profiles of UxlVx and y waves. The results for c = 10 on the other hand exhibits the versus r\ for fixed ft/i(l - (1/c)) and X~' values and various temporal-wave characteristics. Shown in Fig. 11 are some typi c values. The boundary layer separation is found to be delayed cal curves which displays u\H}\ and y versus rj for two values for increasing | and X"'. of c (.64 and 10) at .x = 1.5. In summary, an additional parameter c is required to explain For traveling waves with fixed \"' (not shown here) and .2 the results compared to the spatial and temporal waves, and the < c < 5, the results indicate the following. For X."' = .2 and small and large c results are similar to those for the spatial and low and high c results are similar and consistent with those temporal wave, respectively. for the spatial and temporal waves, respectively, for the same parameter values. The situation is almost similar for waves with X~' = 1 except that the response magnitudes and nonlinearities for the low c values are extreme, i.e., even larger and more complex than those for the spatial wave. Wake centerline velocity responses (not shown here) are more asymmetric and complex compared to the spatial wave. This complexity of the nature of the wake bias is also evident in the boundary layer and wake averaged values (also not shown here) of the PE Eq. (12).
Summary and Conclusions Results are presented for the model problem of a flat-plate boundary layer with temporal-, spatial- and traveling-wave ex ternal flows. For the temporal wave, close agreement with previ ous experimental and computational studies is observed. The magnitude and nature of the response are small and simple such that it is essentially a small disturbance on the steady solution. The results are explainable in terms of the single parameter ^. Boundary-layer separation is delayed for increasing ^. The wake response is smaller in comparison to the boundary-layer re 4.0-, 90 n 1 0.30 t = 0,1 sponse and no wake bias is observed and plug flow is attained 2 0.50 3.0 3 0,77 with faster recovery for increasing ^. 4 10,00 0 ^2.0 For the spatial wave, analogous Stokes-layer behavior is ob -90 1,0- served, but with significantly increased magnitudes and com plex nature of the response such that it cannot be considered a 0 -T-1 -180- , I I I I I I I I I I I I I I 0 1.0 2,0 3.0 4.0 6.0 6.0 7.0 0 1.0 2,0 3.0 4.0 6.0 6.0 7.0 small disturbance on the without-wave solution. Main features 4.0 1 9an0 ^ n are explainable in terms of the two parameters X"' and jc/X. The 3.0 boundary-layer response maximum overshoot values require
?2.0. l/vRe^ scaling. The boundary-layer separation is delayed for increasing X"'. For the wake response, a clear bias is observed. 1,0- This is attributed to nonlinear interaction between the steady and , ,111 first-harmonic velocity components. In the far wake, damped 0 1,0 2.0 3,0 4.0 5,0 6.0 7,0 0 1.0 2,0 3,0 4,0 5.0 6.0 7.0 n n oscillatory plug flow is recovered with faster recovery for in (b) creasing X~'.
Fig. 10 First-harmonic amplitude and phase atx = 1.0: boundary layer, For the traveling wave, an additional parameter, the wave traveling wave, (a) fixed c and variable f, (6) fixed f and variable c (NS speed c is required to explain the results. Poor agreement with solutions shown) the previous experimental and computational study is observed
544 / Vol. 118, SEPTEMBER 1996 Transactions of the ASME for the boundary-layer response magnitude unless the wave E. P. Rood. The calculations were performed on the CRAY speed is adjusted to a lower value. The limiting behavior for supercomputers of the Naval Oceanographic Office and Na small and large wave speeds approached the spatial and tempo tional Aerodynamic Simulation Program. ral waves, respectively. A smooth transition between the two for intermediate values is also observed. The boundary-layer separation is delayed for increasing ^ and \ '. The wake re sponse also showed similar limiting behaviors for small and References large c values, except for the extreme response for small c and Chen, H. C, and Patel, V, C, 1989, "The Flow around Wing-Body Junctions," medium X~'. The wake centerline velocity responses were more Proc. Symposium on Numerical and Physical Aerodynamic Flows, Vol. 4. Chen, H. C, and Patel, V. C, 1990, "Solutions of Reynolds-Averaged Navier- complex compared to the temporal and spatial wave. Stokes Equations for Three-Dimensional Incompressible Flows," Journal of All three flows are of practical importance with regard to a Computational Physics, Vol. 88, No. 2. large variety of applications. The temporal wave problem has Choi, J. E., 1993, "Role of Free-Surface Boundary Layer Conditions and Nonlinearities in Wave/Boundary-Layer and Wake Interaction," Ph.D. thesis, wide-spread applications and most of the study up to this date The University of Iowa. has been in this area. The spatial wave is closely related to ship Choi, ]. E., and Stern, p., 1993, "Solid-Fluid Juncture Boundary Layer and hydrodynamics problems involving surface-piercing bodies and Wake with Waves," Proceedings Sixth International Conference on Numerical 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 associated wavy free-surface effects. Undoubtedly, other similar Ship Hydrodynamics, Iowa City, Iowa. applications exist. The traveling wave problem represents a Graham, M., 1993, private communication. more general version of temporal and spatial wave problems Hanjalic, K., Jakirlic, S., and Hadzic, I., 1993, "Computation of Oscillating Turbulent Flows at Transitional Re-numbers," Turbulent Shear Flows, Vol. 9, J. and has relevance to a host of engineering and technological C. Andre et al, eds.. Springer-Verlag, pp. 324-342. applications such as in helicopter aerodynamics, ship-hydrody Hill, P. G., and Stenning A. H., 1960, "Laminar Boundary Layers in Oscillating namics, propeller-hull interaction, hydroaccoustics, rotor-stator Flow," ASME Journal of Basic Engineering, pp. 593-608. interaction in turbines etc. Previous studies indicate unsteady Horlock, J. 1968, "Fluctuating Lift Forces on a Airfoils moving through Trans lift effects and complex boundary layer and wake responses for verse and Chordwise Gusts," ASME Journal of Basic Engineering, pp. 494- foils in the presence of travelling wave unsteadiness. The pres 500. Lurie, E. H. 1993, "Unsteady Response of a Two-Dimensional Hydrofoil Sub ent model problem provides basic information on oscillatory ject to High Reduced Frequency Gust Loading," M. S. thesis, Dept. of Ocean external-flow pressure gradient effects on boundary layer and Engg., MIT, Cambridge, MA, wake without complexities facilitating the identification of the Paterson, E., and Stern, F., 1993, "Computation of Unsteady Viscous Flow essential features of the three flows. These studies are critical with Application to the MIT Flapping-Foil Experiment," Proceedings Sixth Inter to enhance our understanding of some features which are more national Conference on Numerical Ship Hydrodynamics, Iowa City, Iowa. Patel, M. H., 1975, "On Laininar Boundary Layers in Oscillatory Flow," fundamental in nature such as the role of such outer waves on Proceedings of the Royal Society of London, Series A, Vol. 347, pp. 99-123. the generation and sustenance of the underlying small scale Patel, M. H., 1977, "On Turbulent Boundary Layers in Oscillatory Flow," turbulence and their possible interactions. While some work has Proceedings of the Royal Society of London, Series A, Vol. 353, pp. 121-143. been done on temporal waves, the results of the present study Poling, D., and Telionis, D., 1986, "The Response of Airfoils to Periodic indicates that further efforts are required to determine the nature Disturbances-The Unsteady Kutta Condition," AIAA Journal, Vol. 24, No. 2, pp. of the interaction between the mean and turbulent motions; 193-199. Spalart, P. R., and Baldwin, B. S., 1989, "Direct Simulation of a Turbulent since, it appears that the conclusions found for temporal waves Oscillating Boundary Layer," Turbulent Shear Flows, 6, J. C. Andre et al., eds.. are not necessarily applicable to spatial and traveling waves, at Springer-Verlag, 417-440. least for laminar flows. This is especially true for travelling Stern, F., 1986,' 'Effects of Waves on the Boundary Layer of a Surface-Piercing waves where there is enough evidence that the wave speed has Body," Journal of Ship Research, Vol. 30, pp. 256-274. a major effect on the boundary layer and wake responses and Stern, F., Hwang, W. S., and Jaw, S. Y., 1989, "Effects of Waves on die Boundary Layer of a Surface-Piercing Flat Plate: Experiment and Theory," Jour very little is known regarding the effect of responses on turbu nal of Ship Re.warch, Vol. 33, pp. 63-80. lent quantities. Stern, F., Choi, J. E„ and Hwang, W. S., 1993, "Effects of Waves on the Wake of a Surface-Piercing Flat Plate; Experiment and Theory," Journal of Ship Research, Vol. 37, pp. 102-118. Acknowledgments Stern, F., Kim, H. T., Zhang, D. H., Toda, Y., Kerwin, J., and Jessup, S., 1994, "Computation of Viscous Flow Around Propeller-Body Configurations: Series This research was sponsored by the Office of Naval Research 60 CB = 0.6 Ship Model," Journal of Ship Research, Vol. 38,pp. 137-157. under grant N00014-92-K-1092 under the administration of Dr. Telionis, D. P., 1981, Unsteady Viscous Flow, Springer-Verlag, New York.
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