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In this work, the Kármán–Pohlhausen (KP) momentum-integral approach based on optimized fourth-order (MX4) polynomial approximations of the velocity and temperature profiles is applied to a classical benchmark problem, namely, that of a cylinder in crossflow with a variable pressure gradient. This enables us to extract closed-form expressions for both hydrodynamic and thermal boundary-layer parameters and then compare the newly found solutions to their counterparts obtained using Pohlhausen's cubic (KP3) and quartic (KP4) polynomials. As usual, the farfield around the cylinder is modeled using potential flow theory and the momentum-integral analysis is paired with Walz's empirical expression for the momentum thickness, which is based on a wide collection of experiments. This procedure permits retrieving explicit relations for the pressure-sensitive KP3, KP4, and MX4 velocity profiles across the boundary layer; one also obtains accurate approximations for the pressure distribution around the cylinder as well as an improved prediction of the separation point, namely, to within 0.87% of the actual location. In this process, refined estimates are produced for several characteristic parameters whose distributions are found to be in favorable agreement with experimental measurements and numerical simulations. These include the disturbance, momentum, and displacement thicknesses as well as the skin friction, pressure, and total drag coefficients. Finally, the thermal analysis is undertaken using both isothermal and isoflux boundary conditions. For each of these cases, closed-form analytical solutions are obtained for the local Nusselt number distribution around the cylinder, and these distributions are found to exhibit noticeably reduced errors relative to their classical values.
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On the Kármán momentum-integral approach and the Pohlhausen paradox
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.
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- Award ID(s):
- 1761675
- PAR ID:
- 10589138
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 32
- Issue:
- 12
- ISSN:
- 1070-6631
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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