Abstract
It is shown that an oblique solitary wave may be caused by an oblique strip of enlarged bottom roughness. The analysis is valid for very large Reynolds numbers and very small slopes. Considered are strips with a constant width in the longitudinal direction and with constant roughness. For a strip whose length is much larger than its width, the equations of motion can be simplified by assuming a three-dimensional flow that is independent of the coordinate in the longitudinal direction of the strip. Furthermore, it is assumed that the Froude number in terms of the velocity component normal to the strip is slightly larger than the critical value 1. This facilitates application of an asymptotic expansion method developed previously for plane flow. The analysis does not require modeling of the Reynolds stresses. It turns out that the velocity profile of the three-dimensional flow is not skewed in the first order. The shape, amplitude, and position of the oblique stationary wave are obtained from solutions of the steady-state version of an extended Korteweg-de-Vries equation. In general, the oblique wave is a classical solitary wave with a long, but shallow tail. However, tails are missing for certain combinations of Froude number and parameters characterizing the strip. Measurements and numerical solutions of the full Reynolds equations are already available for orthogonal solitary waves, lending support to the results of the asymptotic analysis. In addition, a novel boundary condition at the free surface is given, and the velocity distribution in fully developed flow is determined.
Highlights
In recent years, oblique solitary waves, including stationary ones, have garnered considerable attention in applied mathematics and fluid mechanics, see Ablowitz and Baldwin (2012), Chakravarty and Kodama (2009; 2014), Chen (1999), Kodoma (2010), Li et al (2011), and Nepomnyashchy and Velarde (1994) for examples and further references
For a strip whose length is much larger than its width, the equations of motion can be simplified by assuming a threedimensional flow that is independent of the coordinate in the longitudinal direction of the strip
It is assumed that the Froude number in terms of the velocity component normal to the strip is slightly larger than the critical value 1
Summary
Oblique solitary waves, including stationary ones, have garnered considerable attention in applied mathematics and fluid mechanics, see Ablowitz and Baldwin (2012), Chakravarty and Kodama (2009; 2014), Chen (1999), Kodoma (2010), Li et al (2011), and Nepomnyashchy and Velarde (1994) for examples and further references. There is the additional difficulty that the unknown free surface has to be determined for nearcritical flow conditions It is, not surprising that stationary solitary waves in turbulent free-surface flow were only found in experiments after they had been predicted theoretically by a suitable analysis; Phys. The analytical results provided the basis for numerical solutions of the Reynolds equations (Stojanovic et al, 2019) These previous investigations concerned solitary waves that are orthogonal to the free stream. In the case of oblique solitary waves, which is the topic of the present paper, the flow is three-dimensional and the angle between the wave and the free stream enters the problem as an additional parameter. In the Appendix, the use of a logarithmic velocity distribution in the defect layer is recommended on the basis of a novel boundary condition at the free surface
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