Classical weighted essentially non-oscillatory (WENO) and other nonlinear schemes often face challenges in achieving steady-state convergence, although substantial progress has already been made in the simulations of compressible Euler equations, there are few contributions for the steady-state simulations of compressible Navier-Stokes (NS) equations. To address this issue, we adopt the fifth-order hybrid WENO (WENO-H) finite difference scheme, designed to achieve machine-zero residual in numerical iterations. The WENO-H scheme employs fifth-order linear reconstruction in smooth regions, guided by an effective smoothness detector, and smoothly degrades to lower-order reconstruction to prevent nonphysical oscillations. It ensures seamless transitions from smooth to discontinuous regions through a smoothing transition zone. In solving compressible NS equations in curved geometries on Cartesian grids, ghost point values outside the physical domain are determined using the fifth-order WENO extrapolation method coupled with simplified inverse Lax-Wendroff procedures. Two sets of ghost point values are utilized to handle convective and diffusive terms discretization near boundaries. Furthermore, the pressure term in primitive variables is substituted with the temperature term to facilitate the imposition of the adiabatic boundary condition. A transitional interpolation technique is proposed to enhance steady-state convergence near free-stream boundaries. Numerical experiments demonstrate that the WENO-H scheme achieves robust steady-state convergence across extensive benchmark examples of NS equations in curved geometries. The scheme exhibits good non-oscillatory property, particularly in scenarios involving strong discontinuities. Moreover, it provides high resolution for both steady and unsteady problems containing multi-scale structures.

Contemporary evaluation and surgical approaches in congenital aortic valve disease have yielded limited success. The ability to evaluate and understand detailed flow characteristics surrounding surgical repair may be beneficial. This study explores the feasibility and utility of echocardiographic-based blood speckle imaging (BSI) in assessing pre- and post-operative flow characteristics in those with non-stenotic congenital aortic root disease undergoing aortic valve repair or valve-sparing root replacement (VSRR) surgery. Transesophageal echocardiogram was performed during the pre-operative and post-operative assessment surrounding aortic surgery for ten patients with non-stenotic congenital aortic root disease. BSI, utilizing block-matching algorithms, enabled detailed visualization and quantification of flow parameters from the echocardiographic data. Post-operative BSI unveiled enhanced hemodynamic patterns, characterized by quantified changes suggestive of the absence of stenosis and no more than trivial regurgitation. Rectification of an asymmetric jet and the reversal of flow on the posterior aspect of the ascending aorta resulted in a reduced oscillatory shear index ((Formula presented.)) of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.), increased peak wall shear stress of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.), and increased time-averaged wall shear stress of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.). This correction potentially attenuates cellular alterations within the endothelium. This study demonstrates that children and young adults with non-stenotic congenital aortic root disease undergoing valve-preserving operations experience significant improvements in flow dynamics within the left ventricular outflow tract and aortic root, accompanied by a reduction in (Formula presented.). These hemodynamic enhancements extend beyond the conventional echocardiographic assessments, offering immediate and valuable insights into the efficacy of surgical interventions.

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.