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This work explores simple relations that follow from the momentum-integral equation absent a pressure gradient. The resulting expressions enable us to relate the boundary-layer characteristics of a velocity profile, u(y), to an assumed flow function and its wall derivative relative to the wall-normal coordinate, y. Consequently, disturbance, displacement, and momentum thicknesses, as well as skin friction and drag coefficients, which are typically evaluated and tabulated in classical monographs, can be readily determined for a given profile, F(ξ) = u/U. Here, ξ = y/δ denotes the boundary-layer coordinate. These expressions are then employed to provide a rational explanation for the 1921 Pohlhausen polynomial paradox, namely, the reason why a quartic representation of the velocity leads to less accurate predictions of the disturbance, displacement, and momentum thicknesses than using cubic or quadratic polynomials. Not only do we identify the factors underlying this behavior but also we proceed to outline a procedure to overcome its manifestation at any order. This enables us to derive optimal piecewise approximations that do not suffer from the particular limitations affecting Pohlhausen’s F = 2ξ − 2ξ3 + ξ4. For example, our alternative profile, F = (5ξ − 3ξ3 + ξ4)/3, leads to an order-of-magnitude improvement in precision when incorporated into the Kármán–Pohlhausen approach in both viscous and thermal analyses. Then, noting the significance of the Blasius constant, s¯≈1.630 398, this approach is extended to construct a set of uniformly valid solutions, including F=1−exp[−s¯ξ(1+12s¯ξ+ξ2)], which continues to hold beyond the boundary-layer edge as y → ∞. Given its substantially reduced error, the latter is shown, through comparisons to other models, to be practically equivalent to the Blasius solution. 

In this work, we generalize the expression of an approximate delta function (ADF), which is a finite- order polynomial that holds identical integral properties to the Dirac delta function, particularly, when used in conjunction with a finite-order polynomial integrand over a finite domain. By focusing on one- dimensional configurations, we show that the use of generalized ADF polynomials can be effective at recovering and extending several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements reduces the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Presently, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to two benchmark cases, namely, the linear wave and nonlinear Burgers’ equations in one-dimensional space. 

Abstract This work extends a sequence of studies devoted to the analysis of the laminar flow in porous channels with retracting walls. This problem was originally used to model slab propellant grain regression. After identifying a subtle endpoint singularity that affects the former solution in its third derivative, a stretched variable is introduced to capture the rapid variations in the channel's core. The core refers to the midsection plane where the shear layer is displaced due to hard blowing at the walls. Then using matched‐asymptotic expansions with logarithmic corrections, a composite solution is developed following successive integrations that start with the fourth derivative. In the process, the inner correction is retrieved from the fourth‐order equation governing the symmetric injection‐driven flow near the core. The resulting approximation is expressed in terms of generalized hypergeometric functions and is confirmed using numerics and limiting process verifications. The composite solution is shown to outperform the former, outer solution, as the core is approached or as the injection Reynolds number is increased. Without undermining the practicality of the former solution outside the thin core region, the development of a matched‐asymptotic approximation enables us to suppress the often overlooked singular terms, thus ensuring a uniformly valid outcome down to the third and fourth derivatives, which affect the pressure distribution and its normal gradients.