Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems

Abstract

Bounds on turbulent averages in shear flows can be derived from the Navier--Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed at each Reynolds number Re by numerically optimizing subject to a spectral constraint, which requires a quadratic integral to be nonnegative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3D) fields. Here we compute optimal bounds over 2.5D fields and then verify, without doing computations over 3D fields, that the bounds indeed apply to 3D flows. One way is to directly check that an optimizer computed using 2.5D fields satisfies the spectral constraint for all 3D fields. A second way uses a criterion we derive that is based on a theorem of Busse (ARMA 47:28, 1972) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3D constraint, is lower computational cost and natural extrapolation of the criterion to large Re. We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow, and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5D fields satisfy our criterion, so they hold for 3D flows. For Couette flow, where bounds have been previously computed using 2.5D fields by Plasting & Kerswell (JFM 477:363, 2003), our criterion holds only up to moderate Re, so at larger Re we directly verify the 3D spectral constraint. Over the Re range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3D flows.

Figures (3)

• Figure 1: (a) Optimal lower bounds on mean dissipation in Waleffe flow, plotted as upper bounds ($\bullet$) on the friction coefficient $\varepsilon$ defined for this model by \ref{['eq: ff waleffe']}, along with the optimal lower bounds ($$) on $\varepsilon$ that take the laminar value $1/\Rey$. (b) Confirmation of the $\chi\le1$ criterion, which implies that the bounds apply to all 3-D flows despite being computed over 2.5-D velocity fields.
• Figure 2: (a) Optimal upper bounds on mean dissipation in Couette flow, computed over 2.5-D velocity fields with no additional constraints ($\bullet$) and with constraints enforcing Busse's criterion ($\circ$). These are plotted as upper bounds on the friction coefficient $\varepsilon$ defined for this model by \ref{['eq: ff couette']}, along with the optimal lower bounds ($$) on $\varepsilon$ that take the laminar value $1/\Rey$. The large-$\Rey$ asymptotes are approximately 0.0086 and 0.0097, respectively. (b) Values of $\chi$ for the bounds computed without enforcing Busse's $\chi\le1$ criterion, which violate the criterion above $\Rey\approx254$.
• Figure 3: (a) Minimum eigenvalues $\lambda_{\min}(j,k)$ of the spectral constraint eigenproblem \ref{['eq: fourier transformed eigenvalue problem']} in the $\Rey=1000$ case, computed by solving \ref{['eq: vorticity/velocity EVP']}. (b) Minimum of $\lambda_{\min}$ over streamwise wavenumbers $k$ at various $\Rey$ for spanwise wavenumbers $j=1,2,3$.