摘要:

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non‐Hamiltonian, withMslow andNfast variables, the corresponding reduced problem is of orderM+ 1; in general it involves all of the slow variables,P1,…,PM, plus the resonant phaseQ. In this paper, the solution of a general non‐Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well‐known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution ofQ. Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak‐Luke, where knowledge of the asymptotic solution forQ(as well as higher‐order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exac

ISSN:0022-2526    

DOI:10.1002/sapm199594283

年代:1995

数据来源: WILEY

摘要:

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by “triad‐like” ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly ment

ISSN:0022-2526    

DOI:10.1002/sapm1995942131

年代:1995

数据来源: WILEY

摘要:

A method is described to determine exact general solutions of the steady two‐dimensional Navier‐Stokes equations. The solutions can also be utilized to construct the stream function for a class of unsteady flows, which in turn contain two arbitrary analytic functions of a complex variable defined in the fluid reg

ISSN:0022-2526    

DOI:10.1002/sapm1995942169

年代:1995

数据来源: WILEY

摘要:

The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter (S) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re2ifk2SRe2is kept constant, wherekis the absolute value of the wavenumber vector. For small values ofSexponential growth of free‐surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flo

ISSN:0022-2526    

DOI:10.1002/sapm1995942183

年代:1995

数据来源: WILEY