16.7 OC680 Assignment 6, Due Wednesday Feb. 28

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Mildred Anderson
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1 292 CHAPTER 16. NOTES Figure 16.30: Profiles of (a) velocity and (b) shear for the separating boundary layer OC680 Assignment 6, Due Wednesday Feb. 28 1: Instability of a separating boundary layer [10] A priori unstable? The profile has an inflection point which represents a shear maximum. Instability will occur if not prevented by viscosity, stratification (not a factor here) or boundaries. Here is my script: % OC680, 2018, script Hmwk6_proj1 % Separating BL problem % W. Smyth, Feb clear close all fs=18; % define parameter values direc= /Users/smyth/Dropbox/HOME/Projects/Book_Instability/figures/ ; Lz=3.0; del=.02; z=[del:del:lz-del] ; N=length(z); U=z.^2.*(6-8*z+3*z.^2); U(z>1)=1; ks=[0:.05:2.1]; Res=10.^linspace(6,1,30); bc= rf ; %

16.7. OC680 ASSIGNMENT 6, DUE WEDNESDAY FEB. 28 293 Figure 16.31: Growth rate versus wavenumber and Reynolds number for the separating boundary layer.

2 16.7. OC680 ASSIGNMENT 6, DUE WEDNESDAY FEB Figure 16.31: Growth rate versus wavenumber and Reynolds number for the separating boundary layer. % plot background profile U* figure subplot(1,2,1) plot(u,z, linewidth,2) xlabel( U*, fontsize,fs, fontangle, italic ) ylabel( z*, fontsize,fs, fontangle, italic ) ylim([0 1.5]) set(gca, fontsize,fs-2) subplot(1,2,2) plot(ddz(z)*u,z, linewidth,2) xlabel( U_z*, fontsize,fs, fontangle, italic ) ylim([0 1.5]) set(gca, fontsize,fs-2) print( -djpeg,[direc hmwk6_1a ]) %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % loop to compute sig(k,re) (takes a while!) % loop over k nk=length(ks) nre=length(res) for i=1:nk k=ks(i); % loop over Re for j=1:nre

3 294 CHAPTER 16. NOTES nu=1/res(j); [s]=vsf(z,u,nu,k,0,bc); sig(i,j)=real(s(1)); end disp([num2str(i/nk) done ]) % track progress end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % Find the critical Re by fitting a quadratic to sig(re) and finding the % zero. [ms,is]=max(sig,[],1); [mms,js]=max(ms); k_mx=ks(is(js)); Re_mx=Res(js); sss=max(real(sig)); jjj=(max(find(sss>max(sss)/1000))) p=polyfit(log10(res(jjj-2:jjj)),sss(jjj-2:jjj),2) Re_root=10.^roots(p) Re_err=abs(Re_root-Res(jjj)); Re_crit=Re_root(Re_err==min(Re_err)) % plot results figure contourf(ks,res,sig,max(sig(:))*[0:.1:.9]); hold on; plot(xlim,280*[1 1], k ) set(gca, yscale, log ) xlabel( k*, fontsize,fs+2, fontangle, italic ) ylabel( Re, fontsize,fs+2, fontangle, italic ) set(gca, fontsize,fs-2) %title( Growth rates for separating boundary layer, fontsize,16) lab=sprintf( Re_{crit}=%.0f,Re_crit) text(.05,.12,lab, units, normal, fontsize,fs-2) lab=sprintf( max \\sigma = %.2e at Re = %.2e, k = %.2f,mms,Re_mx,k_mx) text(.05,.05,lab, units, normal, fontsize,fs-2) colorbar print( -djpeg,[direc hmwk6_1b ]) return If you use fine enough resolution in Re, the minimum value for instability is 280, consistent with the published result. It helps to interpolate, as shown in the code above. If you got a much smaller result, e.g. 140, it is because you have the upper and lower boundaries reversed in your 4th derivative matrix. At the critical Reynolds number, the frozen flow approximation is absolutely not valid, since the growth rate is zero and therefore cannot possibly be fast in comparison to the viscous spreading of the background flow. The growth rate decreases monotonically with decreasing Reynolds number, so this flow is stabilized by viscosity. In other words, the flow is most unstable in the inviscid limit. This indicates that the instability is driven by the inflection point mechanism that applies in inviscid flow (e.g. assignment #2). Viscosity merely modifies the instability by damping its growth rate. This also assures us, via Squire s theorem, that

16.7. OC680 ASSIGNMENT 6, DUE WEDNESDAY FEB. 28 295 Figure 16.32: Echosounder image of instabilities observed in sill flow, from Armi & Farmer, 2002, PRSL.

4 16.7. OC680 ASSIGNMENT 6, DUE WEDNESDAY FEB Figure 16.32: Echosounder image of instabilities observed in sill flow, from Armi & Farmer, 2002, PRSL. Superimposed are estimates of (1) the wavelength of a stratified shear flow instability, (2) the thickness of the separating boundary layer, and (3) the wavelength of what may be a separating boundary layer instability. the 2D modes we have been investigating are indeed the most unstable. The maximum growth rate at Re=106 growth rate is 0.107, which is about 0.06 times the maximum absolute shear. The wavelength of the fastest growing mode is 2p/1.15 = 5.5. The layer thickness, in this scaling, is just 1 (i.e. the length scale is the layer thickness), so the ratio of the wavelength to the layer thickness is 5.5. In the echosounder image, the wavelength of the dominant disturbance downstream of the separation point looks to me to be about 4 times the original thickness of the boundary layer (indicated by lines 1 and 2 on figure 16.32). Multiplying this by 4 to account for the aspect ratio of the image, I get 16, which is of the same order of magnitude as 6. Another way to make this comparison is to draw a line indicating the predicted wavelength (line 3), so that the reader can compare visually with the scales of the observed waves. Trying to guess the velocity profile from an echosounder image is an extremely uncertain business, and this is about as close as I would expect to come. Ideally, one would look at ADCP velocity measurements and make a more careful estimate but, for an order of magnitude comparison, this will do. 2: The half-holmboe instability [10] (a) The algebra is basically identical to section (i) The dispersion relation is c2 = b0. 2k (16.7.1) (ii) The positive and negative values of c represent right- and left-going interfacial gravity waves. (b) The procedure is similar to sections and

5 296 CHAPTER 16. NOTES Figure 16.33: Growth rate versus wavenumber and bulk Richardson number for the half-holmboe problem. Figure 16.34: (a) Phase speed and (b) growth rate versus wavenumber for the half- Holmboe problem.

6 16.7. OC680 ASSIGNMENT 6, DUE WEDNESDAY FEB (i) The dispersion relation is c 2 Ri b c Ri b 2k 2k 4k 2 e 2k = 0. In the limit of large k, the term with the exponential function drops out and the solution describes three waves with phase speeds given by: and 1/2 Rib c = ±, 2k c = 1 1 2k. The first pair are right- and left-going gravity waves riding on the buoyancy interface. Without the scaling, this is equivalent to (16.7.1). The third wave is a vorticity wave as described in section Its phase speed is between zero and 1. As k is decreased from, the right-going gravity wave phase-locks with the shear wave and resonates to form an instability. (ii) See figure (iii) See figure : Instabilities in a plunging downslope flow [20] See figure (a) I get s = at k = 0.47 The growth rate is much greater than 1/Re, so the frozen flow hypothesis is valid. S mx = 1, so the scaled growth rate is In contrast, the boundary layer instability in project #1 gave a scaled growth rate of 0.06, considerably smaller than this. The main difference is the proximity of the lower boundary in project #1, which tends to damp instabilities. (b) I get s = at k = The growth rate is much greater than 1/Re, so the frozen flow hypothesis is valid. The behavior at large k is an artifact of coarse resolution. It drops away as resolution is increased (see comparison below), while behavior near the FGM is not affected. (c) I get s = at k = The growth rate is a factor of ten greater than 1/Re, so the frozen flow hypothesis is marginally valid. Note that decreasing Re has an effect similar to increasing resolution. It smooths the eigenfunctions, making them easier to resolve on a coarse grid. Thus, the anomalous behavior at high k is removed even when D = 0.2. (d) I get s = at k = The growth rate is similar to 1/Re, so the frozen flow hypothesis is not valid. A typical wavenumber from cases a-c is k = 0.45, which yields a wavelength of 14h, or 7 times the thickness of the shear layer (2h). Correcting for the aspect ratio of the image (4), this wavelength would appear about 7/4 times the thickness of the shear layer (figure 16.32). On the figure, the vertical line indicates my estimate of the initial thickness of the shear layer. The horizontal line is longer by a factor 7/4. This actually agrees pretty well with the typical scale of the earliest deformations of the shear layer (which may be pure luck!) Further downstream, larger structures emerge, possibly as a result of merging of the primary billows.

7 298 CHAPTER 16. NOTES Figure 16.35: Growth rate versus wavenumber for four cases of a stratified shear flow. U = tanhz, B = Ri b tanhz. Impermeable, fixed-buoyancy boundaries at z = ±4. Grid spacing D = 0.2. Note under-resolution and boundary effects at high and low k.

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