A numerical method for treating strongly interactive three-dimensional viscous—inviscid flows

Receptivity of a laminar boundary layer to the interaction of time-harmonic free-stream disturbances with a three-dimensional roughness element is studied. The three-dimensional nonlinear triple–deck equations are solved numerically to provide the basic steady-state motion. At high Reynolds numbers, the governing equations for the unsteady motion are the unsteady linearized three-dimensional triple-deck equations. These equations can only be solved numerically. In the absence of any roughness element, the free-stream disturbances, to the first order, produce the classical Stokes flow, in the thin Stokes layer near the wall (on the order of our lower deck). However, with the introduction of a small three-dimensional roughness element, the interaction between the hump and the Stokes flow introduces a spectrum of all spatial disturbances inside the boundary layer. For supercritical values of the scaled Strouhal number, S0 > 2, these Tollmien–Schlichting waves are amplified in a wedge-shaped region, 15° to 18° to the basic-flow direction, extending downstream of the hump. The amplification rate approaches a value slightly higher than that of two-dimensional Tollmien–Schlichting waves, as calculated by the linearized analysis, far downstream of the roughness element.

Section: Steady-state Flowmentioning

confidence: 99%

Receptivity of a laminar boundary layer to the interaction of a three-dimensional roughness element with time-harmonic free-stream disturbances

Tadjfar

Bodonyi

1992

“…The grid for the streamwise variable X is the same as that for the lower-deck boundary-layer equations; the grid in the wall-normal direction is of course different. The numerical method is the same as described in Bodonyi & Duck (1988), where further details can be found.…”

Section: Discussionmentioning

confidence: 99%

“…Appendix C. The method of solving numerically the Laplace equation for the pressure An alternative approach to handle the divergence of the integral in the P-D law is to solve numerically the pressure equation in the upper deck simultaneously with the boundary-layer equations in the lower deck, as was suggested by Bodonyi & Duck (1988). In terms of X and the wall-normal coordinate for the upper deck (Smith 1989) y = λ 5/4 (1 − M 2 ) 7/8 C −3/8 (T w /T ∞ ) −3/2 y * /( 3 L), ( The Laplace equation (C 2) for the pressure is solved in the domain: X 0 X X I and 0 ȳ ȳ K .…”

Section: Discussionmentioning

confidence: 99%

“…For the supercritical case of our interest, the growth rate −α i > 0 and hence the displacement functionÃ tends to infinity as X → ∞ (see (2.24)), which makes the integral in the P-D law (2.20) divergent, as was pointed out by Bodonyi & Duck (1988). The integral must be interpreted as the finite part in the sense of Hadamard, which requires careful numerical evaluation; see Appendix A.…”

Section: Governing Equations For the Scattered Perturbation Fieldmentioning

confidence: 99%

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A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue

Dong

2016

This paper is concerned with the rather broad issue of the impact of abrupt changes (such as isolated roughness, gaps and local suctions) on boundary-layer transition. To fix the idea, we consider the influence of a two-dimensional localized hump (or indentation) on an oncoming Tollmien-Schlichting (T-S) wave. We show that when the length scale of the former is comparable with the characteristic wavelength of the latter, the key physical mechanism to affect transition is through scattering of T-S waves by the roughness-induced mean-flow distortion. An appropriate mathematical theory, consisting of the boundary-value problem governing the local scattering, is formulated based on triple deck formalism. The transmission coefficient, defined as the ratio of the amplitude of the T-S wave downstream the roughness to that upstream, serves to characterize the impact on transition. The transmission coefficient appears as the eigenvalue of the discretized boundary-value problem. The latter is solved numerically, and the dependence of the eigenvalue on the height and width of the roughness and the frequency of the T-S wave is investigated. For a roughness element without causing separation, the transmission coefficient is found to be about 1.5 for typical frequencies, indicating a moderate but appreciable destabilizing effect. For a roughness causing incipient separation, the transmission coefficient can be as large as O(10), suggesting that immediate transition may take place at the roughness site. A roughness element with a fixed height produces the strongest impact when its width is comparable with the T-S wavelength, in which case the traditional linear stability theory is invalid. The latter however holds approximately when the roughness width is sufficiently large. By studying the two-hump case, a criterion when two roughness elements can be regarded as being isolated is suggested. The transmission coefficient can be converted to an equivalent N-factor increment, by making use of which the e N -method can be extended to predict transition in the presence of multiple roughness elements.

“…Smith (1983) (extended by Bodonyi and Duck 1988) presented a finite-difference scheme for treating flows of this type, using a system of rotated coordinates, enabling the problem to be made quasi -two-dimensional. …”

Section: (1980)mentioning

confidence: 99%

Three-dimensional marginal separation

Duck

1989

SUMMARYThe three-dimensional marginal separation of a boundary layer along a line of symmetry is considered. The key equation governing the displacement function is derived, and found to be a non-linear integral equation in two space variables. This is solved iteratively using a pseudo -spectral approach, based partly in double Fourier space, and partly in physical space.Qualitatively the results are similar to previously reported two-dimensional results (which are also computed to test the accuracy of the numerical scheme) ; however quantitatively the three -dimensional results are much different.