Andrei A. Kolyshkin

Stability analysis of shallow wake flows

Experimentally observed periodic structures in shallow (i.e. bounded) wake flows are believed to appear as a result of hydrodynamic instability. Previously published studies used linear stability analysis under the rigid-lid assumption to investigate the onset of instability of wakes in shallow water flows. The objectives of this paper are: (i) to provide a preliminary assessment of the accuracy of the rigid-lid assumption; (ii) to investigate the influence of the shape of the base flow profile on the stability characteristics; (iii) to formulate the weakly nonlinear stability problem fo rs hallow wake flows and show that the evolution of the instability is governed by the Ginzburg–Landau equation; and (iv) to establish the connection between weakly nonlinear analysis and the observed flow patterns in shallow wake flows which are reported in the literature. It is found that the relative error in determining the critical value of the shallow wake stability parameter induced by the rigid-lid assumption is below 10% for the practical range of Froude number. In addition, it is shown that the shape of the velocity profile has a large influence on the stability characteristics of shallow wakes. Starting from the rigid-lid shallow-water equations and using the method of multiple scales, an amplitude evolution equation for the most unstable mode is derived. The resulting equation has complex coefficients and is of Ginzburg– Landau type. An example calculation of the complex coefficients of the Ginzburg– Landau equation confirms the existence of a finite equilibrium amplitude, where the unstable mode evolves with time into a limit-cycle oscillation. This is consistent with flow patterns observed by Ingram & Chu (1987), Chen & Jirka (1995), Balachandar et al .( 1999), and Balachandar & Tachie (2001). Reasonable agreement is found between the saturation amplitude obtained from the Ginzburg–Landau equation under some simplifying assumptions and the numerical data of Grubi˘´ c et al .( 1995). Such consistency provides further evidence that experimentally observed structures in shallow wake flows may be described by the nonlinear Ginzburg–Landau equation. Previous works have found similar consistency between the Ginzburg–Landau model and experimental data for the case of deep (i.e. unbounded) wake flows. However, it must be emphasized that much more information is required to confirm the appropriateness of the Ginzburg–Landau equation in describing shallow wake flows.

Perturbation dynamics in unsteady pipe flows

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

Can the dynamics of shallow wakes be reproduced from a single time-averaged profile?

To date, all linear stability analyses of shallow wakes are performed on a series of wake-type profiles, but the wake generating object and its associated separation points are always neglected in such analyses. The results of numerical modeling of depth-averaged shallow water equations are presented in order to assess the importance of a wake generating body (cylinder) for the vortex structure and the global frequency of oscillations. More specifically, this Brief Communication seeks to answer the following question: is it possible to reproduce the dynamics of a vortex street in shallow wakes by only knowing the time-averaged profile at a particular location downstream of the bluff body? This is accomplished by comparing the wake characteristics for the case with a cylinder situated in the middle of the flow domain with the flow characteristics for the case without a cylinder. The upstream boundary condition for the no cylinder case consists of a wake-type velocity profile. It is found that spatio-tempor...

Investigation of the mechanisms responsible for the breakdown of axisymmetry in pipe transient

Existing transient pipe flow models assume that the flow field is stable and axisymmetric. Recent experimental and theoretical researches show that water hammer flows can become unstable and these instabilities lead to the breakdown of the axisymmetric assumption and to significant changes to the flow held and the magnitude and distribution of wall shear stresses. In addition, previous experiments revealed that in some cases the flow asymmetry developed in a time scale smaller than the water hammer time scale (ratio of pipe length to wavespeed) and in other cases the instability developed in a time scale of the order of the water hammer time scale. The current stability analysis is consistent with previous experimental observations. In particular, it is shown that there are two distinct mechanisms of instability, namely, a transient growth mechanism which develops in time scale shorter than the water hammer time scale and an exponential growth mechanism which develops in time scale of the order of the water hammer time scale; and the instability is indeed of asymmetric nature. Perturbations with large streamwise length scale (i.e. large streamwise wavelength) are more susceptible to the transient instability while perturbations with small streamwise length scale are more susceptible to exponential instability. The exponential instability may become active only if ihe flow “survives” the instability associated with the transient growth. The transient flow instability (sometimes referred to as bypass instability) can arise even when the flow is deemed to be stable by the traditional modal stability analysis. Such instability is known to occur in other Hows and lands support to the experimental finding that instabilities can occur shortly after the passage of the first water hammer wave. Physically. Ihe bypass instability is associated with the tilting of base flow vortices by the velocity fluctuations. Plots of the flow lield clearly show the breakdown of the axisymmetric assumption and produce asymmetric velocity profiles consistent with those observed in water hammer experiments.

Mathematical model for eddy current testing of metal plates with two cylindrical flaws

Semi-analytical solution of eddy current testing problem is presented in the paper. An excitation air-core coil is located above a conducting plate with two flaws in the form of cylindrical holes located at top and bottom parts of the plate. The change in impedance of the coil is computed for different frequencies of the excitation current. Potential applications (analysis of the effect of corrosion, estimation of thickness and conductivity of metal coatings and quality testing of spot welding) are discussed.

SPATIAL STABILITY ANALYSIS OF SHALLOW MIXING LAYERS WITH VARIABLE FRICTION COEFFICIENT

Spatial linear stability analysis of shallow mixing layers is performed in the present paper for the case where friction force varies considerably in the transverse direction. Floodplains with vegetation give practical examples of such a situation. Linear stability problem is formulated for the case where friction coefficient is a function of the transverse coordinate. Numerical results show that growth rates for the case of variable friction are larger than for the case of constant friction.

Linear and Weakly Nonlinear Instability of Shallow Mixing Layers with Variable Friction

Linear and weakly nonlinear instability of shallow mixing layers is analysed in the present paper. It is assumed that the resistance force varies in the transverse direction. Linear stability problem is solved numerically using collocation method. It is shown that the increase in the ratio of the friction coefficients in the main channel to that in the floodplain has a stabilizing influence on the flow. The amplitude evolution equation for the most unstable mode (the complex Ginzburg–Landau equation) is derived from the shallow water equations under the rigid-lid assumption. Results of numerical calculations are presented.

Eddy Current Testing Models for the Analysis of Corrosion Effects in Metal Plates

Direct method for the solution of eddy current testing problem for the case where an air core coil is located above a conducting two-layer plate with a flaw in the form of a cylindrical inclusion with reduced electrical conductivity is presented in the paper. Semi-analytical approach (the TREE method) is used to construct the solution of the system of equations for the components of the vector potential. The flaw is assumed to be symmetric with respect to the coil. Numerical calculations are performed using the proposed model and Comsol Multiphysics software. The obtained values of the change in impedance of the coil for both methods are found to be in a good agreement. The proposed model can be used for the assessment of the effect of corrosion in metal plates.

DIRECT EDDY CURRENT METHOD FOR VOLUMETRIC FLAWS OF CYLINDRICAL SHAPE

Abstract. A quasi-analytical method for the solution of direct eddy current testing problems for the case of cylindrical volumetric flaws is presented in the paper. The method is based on a simple physical assumption that the electromagnetic field induced by a coil carrying alternating current is exactly equal to zero at a sufficiently large radial distance from the coil. The axis of the coil concides with the axis of a cylindrical flaw. The method of truncated eigenfunction expansions is used to compute the change in impedance of the coil. Complex eigenvalues are computed numerically using the method which does not require initial approximation for the eigenvalue. Computations are presented for different values of the parameters of the problem. Calculated change in impedance is compared with numerical results obtained by means of Comsol Multiphysics software. Good agreement between quasi-analytical method and numerical solution is found.

Eddy Current Model for Nondestructive Testing of Electrically Conducting Materials with Cylindrical Symmetry

Eddy current method is widely used in practice for quality testing of conducting materials (examples include determination of electrical conductivity, thickness of metal coatings, identification of flaws in a conducting medium). In the present paper a semi-analytical method for solution of direct eddy current problems for the case of a conducting medium of finite size is considered. The method is applied to several eddy current problems with cylindrical symmetry. The following problem is analyzed in detail. Consider a coil with alternating current located above a conducting medium in the form of a circular cylinder (such a model can be used for design of coin validators which are based on the estimation of electrical conductivity of a coin). We assume that the electromagnetic field is exactly zero at a sufficiently large distance from the coil (the distance can be chosen on the basis of the required accuracy of the solution). The solution is constructed using the method of separation of variables which includes two steps where numerical calculations are necessary: (a) computation of complex eigenvalues without good initial guess for the roots and (b) solution of a system of linear algebraic equations. Computations of the change in impedance of the coil for different frequencies with the semi-analytical method are in good agreement with experimental data and results of numerical simulation with finite element method. Solution of other problems with cylindrical symmetry is also discussed (a flaw in the form of a circular cylinder in a conducting half-space or a plate). Such models can be used for the analysis of quality of spot welding (in case of a volumetric flaw) and estimation of the effect of corrosion (for surface flaws).