Plane channel flow transition experiment trans2

Plane channel flow transition experiment trans2 Markus Uhlmann Potsdam Institut f¨ ur Klimafolgenforschung, D-14412 Potsdam [email protected] (14 January 2002)

1. Transition We integrate the incompressible Navier-Stokes equations in time from an initially laminar Poiseuille profile to which we add the most unstable two-dimensional eigenmode obtained from linear stability analysis. At the outer flow Reynolds number Re = h U0 /ν = 8000 (where h is the channel half width, U0 the maximum mean velocity and ν the kinematic viscosity) there is a small band of linearly unstable wavenumbers (cf. figure 1) from which we choose α h = 1, having an associated eigenmode as shown in figure 2. From this eigenmode we can reconstruct a perturbation field in terms of our independent variables ωy (which is not affected due to the two-dimensionality of the perturbation) and ϕ = ∇2 v. The amplitude of the linear perturbation is set such that u0 /U0 ∼ 10−2 initially. In order to provide for a three-dimensional background noise, randomly-phased perturbations with an intensity of O(10−5 ) are added to both ωy and ϕ (cf. the listing in appendix A for the detailled procedure). We have chosen a domain size of Lx = 2πh (streamwise) and Lz = 0.83h (spanwise). Figure 3 shows the time evolution of the plane-averaged skin friction during the present simulation. At around t = 130h/U0 transition sets in accompanied by a sixfold increase in wall friction. At the same time, perturbations of velocity (figure 4) and vorticity (figure 5) undergo a large amplification and a later relaxation towards typical turbulent levels. Furthermore, figures 4 and 5 show that three-dimensionality of the flow sets in around t = 100h/U0 (i.e. w0 , ωx0 and ωy0 take on finite values). Figure 6 shows plane-averaged streamwise velocity profiles at different times. The initial parabolic shape is only slightly perturbed (particularly near the walls) halfway through the transition process (t = 133h/U0) whereas a clearly turbulent profile is established at the latest time of the simulation (t = 274h/U0). In terms of resources, the present experiment – using 96 × 97 × 32 modes in the streamwise, wall-normal and spanwise direction – uses 3MB of core memory per processor and executes over 14h wall clock time on an IBM SP2 (thin nodes 66 MHz) using 24 processors.

2. Statistics of the fully turbulent flow in a narrow box After the above described transition period (t ≈ 280h/U0, cf. figure 3) we continue the simulation for around 30 outer flow time units at an intermediate Reynolds number Re = 5500 and then decrease it to the final Re = 3750. In this regime, the resolution is sufficient for a realistic representation of all relevant scales (i.e. after de-aliasing: ∆x + = + 15, ∆z + = 6, ∆ymax = 5; cf. also Kim et al. (1987)). After a transient of 140 time units, we go on accumulating statistics over a duration of 440 units. The friction-velocity-based

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Reynolds number amounts to Rτ ≈ 155 in this case. Figure 7 shows that the logarithmic region is not realistically represented by a box of this spanwise extent (L+ z ≈ 130; cf. also Jim´enez & Moin (1991)). Near the wall, the flow is, however, very similar to a generic turbulent channel as can be seen from the rms values of the velocity and vorticity fluctuations (figure 8). REFERENCES Jim´ enez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240. Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in a fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.

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Plane channel flow transition experiment trans2 0.01 0 -0.01 -0.02

=(c)/U0 αh-0.03 -0.04 -0.05 -0.06 -0.07 0

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αh Figure 1. Growth rate of normal mode perturbations as a function of streamwise wavenumber α at Re = 8000 (laminar plane channel flow).

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x/h Figure 2. Isolines of the perturbation stream-function of the most unstable mode of laminar plane channel flow at Re = 8000 (streamwise wavenumber of the perturbation α = 1/h).

Appendix A. Generation of the initial field subroutine perturb2(vor,phi,j1,j2) c c c c c c

assigns a perturbation to omy & phi: interpolate Psi (streamfunction) eigenmode from linear stability analysis onto current y-grid and deduce corresponding phi-variation (2d perturbation, cst in z) omega_y does not have perturbation (all 2d)

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t U0 /h Figure 3. Evolution of the plane-averaged wall-friction of case trans2 at Re = 8000. , lower wall. top wall;

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t U0 /h Figure 4. Evolution of the box-averaged velocity fluctuations of case trans2 at Re = 8000. , streamwise; , wall-normal; , spanwise component.

c implicit real*8(a-h,o-z) include "ctes3Dp" complex*16 vor(0:mx1,0:mz1,j1:j2),phi(0:mx1,0:mz1,j1:j2),rii real*4 simple,simp2(2) dimension psir(my),psii(my),ddpsir(my),ddpsii(my),toto(my) parameter(amplin=5.d-3) parameter(ampvor=1.d-5,ampphi=1.d-5) common /fscom/ y(my)

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t U0 /h Figure 5. Evolution of the box-averaged vorticity fluctuations of case trans2 at Re = 8000. , streamwise; , wall-normal; , spanwise component.

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y/h Figure 6. Plane-averaged streamwise velocity profiles during the transition experiment at Re = 8000: , t U0 /h = 0; , t U0 /h = 133; , t U0 /h = 274.

common /fis/ Re,alp,bet,ytop common /wave/ xalp(0:mx1),xbet(0:mz1),iax(mx),ibx(0:mz1), . icx(0:mz1) c write(*,*)’PERTURB_2’,j1,j2 pi=4.d0*datan(1.d0) rii=dcmplx(0.d0,1.d0) c c

/* interpolate eigenvector upon current grid */

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y+ Figure 7. Temporally and plane-averaged streamwise velocity profiles; lines show the present case trans2 at Re = 3750 and lines with symbols correspond to the reference case por16 which was computed at a Reynolds number Re = 3250 and in a larger box of Lx = 8.17h and Lz = πh.

call interpolate(y,psir,psii,my) c

/* pass psi to chebychev space */ call tcheb(psir,1,1,1,-1) call tcheb(psii,1,1,1,-1)

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/* derive in y (need to reconstruct phi=laplace.v; om_y=0) */ call dcopy(my,psir,1,ddpsir,1) call dcopy(my,psii,1,ddpsii,1) call chebd1(ddpsir,1,my,1) call chebd1(ddpsir,1,my,1)

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y/h Figure 8. rms values of velocity and vorticity fluctuations (refer to figure 6 for symbols) at Re = 3750: (a) , u0 ; , v0 ; , w0 ; (b) , ωx0 ; , ωy0 ; , ωz0 .

call chebd1(ddpsii,1,my,1) call chebd1(ddpsii,1,my,1) c c

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/* assign perturbation in f-f-cheb space */ simple=random(simp2,1) angle=dble(simple)*2.d0*pi /* acutally use constant angle so that all slices have the same */ angle=0.d0 kk=0 ii=1

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M. Uhlmann write(*,*)’wavenumber/-length of perturbation: ’, alp*dfloat(ii),2.d0*pi/(alp*dfloat(ii)) do jj=j1,j2 phi(ii,kk,jj)=amplin*cdexp(-rii*angle)* $ (dcmplx(psir(jj),psii(jj))*rii*xalp(ii)**3$ dcmplx(ddpsir(jj),ddpsii(jj))*rii*xalp(ii)) enddo /* add low-level backbround noise at all wavenumbers */ do kk=1,mz1 do ii=1,mx1 do jj=j1,j2 simple=random(simp2,1) angle=dble(simple)*2.d0*pi phi(ii,kk,jj)=ampphi*cdexp(-rii*angle) enddo enddo enddo return end $

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