SUMMARYA discrete singular convolution (DSC) solver is developed for treating incompressible ows. Three di erent two-dimensional benchmark problems, the Taylor problem, the driven cavity ow, and a periodic shear layer ow, are utilized to test the accuracy, to explore the reliability and to demonstrate the e ciency of the present approach. Solution of extremely high accuracy is attained in the analytically solvable Taylor problem. The results of treating the other problems are in excellent agreement with those in the literature.

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$ ), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ( $\beta \gtrapprox 0.785$ ), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$ . The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$ , the Landau coefficient scales with the inverse of the Weissenberg number ( $Wi$ ) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

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SUMMARYA benchmark quality solution is presented for ow in a staggered double lid driven cavity obtained by using the wavelet-based discrete singular convolution (DSC). The proposed wavelet based algorithm combines local methods' exibility and global methods' accuracy, and hence, is a promising approach for achieving the high accuracy solution of the Navier-Stokes equations. Block structured grids with pseudo-overlapping subdomains are employed in the present simulation. A third order RungeKutta scheme is used for the temporal discretization. Quantitative results are presented, apart from the qualitative uid ow patterns. The prevalence of rich features of ow morphology, such as two primary vortex patterns, merged single primary vortex patterns, and secondary eddies, makes this problem very attractive and interesting. The problem is quite challenging for the possible existence of numerically induced asymmetric ow patterns and elliptic instability. Important computational issues like consistence, convergence and reliability of the numerical scheme are examined. The DSC algorithm is tested on the single lid driven cavity ow and the Taylor problem with a closed form solution. The double lid driven cavity simulations are cross-validated with the standard second order ÿnite volume method.

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