Perturbation Stream Function Research Articles

Theoretical analysis and numerical modeling of the evolution mechanism of polymer melt unstable flow in two‐dimensional ultrasonic‐assisted micro‐injection molding

AbstractMicro polymer parts are usually fabricated by ultrasonic‐assisted micro‐injection molding which is an effective and low‐cost method. This method solves some injection molding defects such as short shot by reducing the melt viscosity, and the clarification of its mechanism is a challenge. This article is aimed at studying the evolution mechanism of two‐dimensional ultrasonic vibration on the polymer melt unstable flow. A two‐dimensional ultrasonic vibration, whose displacements components in the‐ and‐directions are large, is applied to the polymer melt. The unstable flow is a flow field with perturbations. The formulas of perturbation stream function and velocity perturbations are established based on the theories of absolute instability and convective instability. The established formulas are verified by numerical analysis. The results show that the flow of polymer melt is convectively unstable and the vortices are formed under the combined actions of shear stress and transverse pressure. The distribution of vortices is bound to affect the mold filling quality. Accordingly, a two‐dimensional ultrasonic vibration with the phase difference of 90° is adopted to make strong vortices occur symmetrically, to improve the uniformity of viscosity field, and to obtain a better melt filling capability during injection molding.

Nonlinear phase dynamics of ideal kink mode in the presence of shear flow

We investigate nonlinear phase dynamics of an ideal kink mode, induced by E × B flow. Here the phase is the cross phase (θ c) between perturbed stream function of velocity () and magnetic field (), i.e. θ c = θ ϕ − θ ψ . A dimensionless parameter, analogous to the Richardson number, (γ kink: the normalized growth rate of the pure kink mode; : normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear. When R i > 1, θ c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R i ≤ 1, θ c enters a phase slipping or oscillating state, corresponding to a nonmodal solution. The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.

Solvability of a supercritical free surface flow problem under gravity-capillarity

In this paper, we consider a problem of a supercritical free surface flow over an obstacle lying on the bottom of a channel in 2D. The flow is irrotational, stationary and the fluid is ideal and incompressible. We take into account both the gravity and the effects of the superficial tension. The problem is nonlinear, it is formulated by the Laplace operator and the dynamic condition defined on the free surface of the fluid domain (Bernoulli equation). Using the perturbation stream function, we linearize the problem and we give a priori properties of the solution. These a priori properties allow us to construct a space where we can use the Lax–Milgram’s theorem to prove the existence and the uniqueness of the solution of the problem.

Non-similar solution of the forced convection of laminar gaseous slip flow over a flat plate with viscous dissipation: linear stability analysis for local similar solution

Forced convection of laminar nearly incompressible gaseous slip flow over an isothermal flat plate at low Mach number with viscous dissipation is considered. The non-similar solutions of hydrodynamical and thermal boundary layers equations with velocity-slip and temperature-jump at the wall are obtained numerically by using the implicit finite difference method. The effects of the modified boundary layer Knudsen number, i.e., the slip parameter and the Eckert number on the heat transfer characteristics are presented graphically and discussed. The numerical results show that for small Eckert number, the slip parameter does not have significant effect on the local heat transfer in the continuum and in slip flow regimes while for the large Eckert numbers, its effect depends that the plate being colder or warmer than the free stream. In addition, we develop a linear stability analysis, based on the traditional normal-mode approach, by assuming local parallel flow approximation, to study the effect of slip parameter on the stability of local similar solution. This approach leads to the usual Orr–Sommerfeld equation which governs the perturbation stream function satisfying slip boundary condition. This equation is solved numerically by using a powerful method based on spectral Chebyshev collocation. For no slip flow, the results for the eigenvalues and the corresponding wave numbers are found in excellent agreement with previous available numerical calculations that supports the validity of our results. Furthermore, the neutral curves of stability in the Reynolds-wave number plane are obtained, for the first time, for the boundary layer in the slip flow regime. The results show that the effect of slip parameter is to increase the critical Reynolds numbers for instability and to decrease the most unstable wave numbers. It is concluded that the rarefaction has a stabilizing effect on the Blasius flow and suggests that the transition to turbulence could be delayed in the slip flow regime.

Sensitivity of African easterly waves to boundary layer conditions

Abstract. A linearized version of the quasi-geostrophic model (QGM) with an explicit Ekman layer and observed static stability parameter and profile of the African easterly jet (AEJ), is used to study the instability properties of the environment of the West African wave disturbances. It is found that the growth rate, the propagation velocity and the structure of the African easterly waves (AEW) can be well simulated. Two different lower boundary conditions are applied. One assumes a lack of vertical gradient of perturbation stream function and the other assumes zero wind perturbation at the surface. The first case gives more realistic results since in the absence of horizontal diffusion, growth rate, phase speed and period have values of 0.5 day−1, 10.83 m s−1 and 3.1 day, respectively. The zero wind perturbation at the surface case leads to values of these parameters that are 50 percent lower. The analysis of the sensitivity to diffusion shows that the magnitude of the growth rate decreases with this parameter. Modelled total relative vorticity has its low level maximum around 900 hPa under no-slip, and 700 hPa under free slip condition.

On the plane problem of the flow around a submerged beam

We consider the problem of the steady flow of an ideal heavy fluid around a submerged beam. The problem is obtained from the free-boundary problem of the flow past a submerged obstacle in the limit of bodies of vanishing thickness. We introduce a special Sobolev space formulation of the problem in term of a perturbed stream function and prove its unique solvability for every value of the unperturbed flow velocity, with the possible exception of a discrete set depending on the geometry of the domain. The asymptotic properties of the solutions are discussed.

Uniqueness and trapped modes in the linear problem of the steady flow over a submerged hollow

We consider the linear problem of the steady flow past a submerged hollow of rectangular shape. The fluid is assumed to be inviscid and incompressible and the problem is posed in term of a perturbed stream function vanishing at infinity. By introducing a suitable variational form of the problem, we discuss unique solvability in the case of a supercritical stream at infinity and find sufficient conditions on the hollow’s size for the existence of non trivial solutions of the homogenous problem (trapped modes).

Linear stability of viscoelastic cone–plate flow in a bounded domain

We analyze the elastic instability of torsional flow of a viscoelastic fluid in a cone-and-plate device with gap angle α. Assuming that the flow is axisymmetric and inertialess, we derive a model eigenvalue problem for the perturbation stream function valid for α≪1. The problem reduces to a singularly perturbed fourth-order boundary value problem related to the biharmonic operator. This problem is solved numerically and the results are compared with those obtained for radially unbounded domains. Assuming that the meniscus perturbation is small enough to preclude capillary instabilities, the flow is found to be slightly more stable than is predicted by the results for radially unbounded regions.

Diagnosis of Optimal Perturbation Evolution in the Eady Model

Abstract The structure and evolution of Eady model singular vector (SV, also referred to as optimal perturbation) streamfunction perturbations are described using a combination of two different partitions of the vector subspace describing all possible streamfunction perturbations. A modal partitioning of the SV perturbation streamfunction (expressing the SV streamfunction as a linear combination of modal structures) is used to ascribe the roles and relative importance of the continuum modes (CMs) and the discrete normal modes (NMs) in SV initial structure and subsequent evolution. In addition, a potential vorticity (PV) partitioning of the SV perturbation streamfunction into parts attributed to the SV PV and the SV boundary thermal anomalies (BTAs) is employed. The structures of the CMs and NMs are described in terms of their characteristic perturbation PV and BTAs. Modal decomposition of the SVs reveals that for all zonal wavenumbers (k), the NMs have the largest projection coefficients (with magnitudes ...

Linear Stability of Modons on a Sphere

The linear stability of two stationary dipolar modon solutions of the nondivergent barotropic vorticity equation on a rotating sphere is investigated. A numerical normal mode analysis of the linearized equation is performed by solving the eigenvalue problem in a spectral model. The modons are models of observed vortical structures in the atmosphere such as planetary waves and atmospheric blocks. The wavelike modon in an eastward zonal flow represents the class of partially localized vortical waves with phase velocities inside the Rossby wave regime. The localized modon in a westward zonal flow represents the class of desingularized point vortex pairs with phase velocities outside the Rossby wave regime. The research into the stability properties of these modons may lead to a deeper insight into the decay and persistency of planetary waves and atmospheric blocks and thereby into the low-frequency variability of the atmosphere. The convergence of the modes with resolution is studied with the matrix method for each resolution T10–T100 and with the iterative power method for every five resolutions T80–T170 and for T341. The modes are tracked through the resolutions using correlations between eigenvectors at subsequent resolutions. The four most unstable modes of the wavelike modon are convergent over T22. For the localized modon, the most unstable mode is convergent over T66, whereas other modes appear to converge to zero growth rate over T100. Both modons are unstable with an e-folding time of several days. The structure of the modes is studied in detail for resolution T85. The most unstable modes have their largest local amplitudes in regions with the strongest gradients of potential vorticity. The modes propagate around the amphidromic points, local centers of revolution, which characterize the topological structure of the modes. The most unstable mode of the wavelike modon has a tripolar structure and propagates unhindered across the boundary circle between the inner and outer region of the modon. The most unstable mode of the localized modon has a quadrupolar structure and propagates north and south of the boundary circle, which is a critical line and an impenetrable barrier. The modons are representatives of two different classes of vortical structures and their quite distinct mode structures can be understood using the refractive index interpretation. The Eliassen–Palm theorem applied to modes as perturbations on modons leads to a condition on the mean spectral wavenumber for an unstable mode: the barotropic wedge of instability in the diagram of perturbation relative vorticity versus perturbation streamfunction. A causal argument regarding the time required for an instability to radiate propagating waves leads to the phase speed condition for a global propagating mode: the spatial extent of the mode is related to the ratio of its e-folding time and oscillation period. These theoretical results are supported by high-resolution numerical results: wavelike modons have localized (trapped) modes, whereas localized modons have all but one wavelike (radiating) mode. The localized modon has a single isolated unstable mode with an additional continuum of propagating neutral modes. For both modons, the most unstable mode propagates eastward between the cells in the inner region and westward along the boundary circle in the outer region.

A Simple Example of Galilean Invariance in the Omega Equation

Abstract This paper is pedagogically motivated to qualitatively demonstrate basic concepts related to Galilean invariance and partial cancellation in the individual forcing terms in the traditional form of the omega equation. The analysis provides examples of the vertical distribution of the primary quasigeostrophic (QG) forcing, showing how the individual forcing terms vary in different coordinate systems while their sum remains constant. The QG forcing is described analytically using an unstable Eady wave solution, which allows depictions of QG forcing similar to those in basic atmospheric dynamics texts. The perturbation streamfunction, temperature, and vertical velocity are seen to be invariant under change of horizontal coordinate system, while the individual magnitudes of the QG vertical motion forcing are not. The total QG forcing remains invariant in all cases and is equal to the QG forcing found from the divergence of the Q vector. The figures provided can supplement those used for traditional st...

On the evolution of near-singular modes of the Bickley jet

The linear stability spectrum of the Bickley jet has neutral modes which have a phase velocity equal to the maximum jet velocity. Unlike critical levels in monotonic shear flows, the stream function associated with these modes is algebraically singular at the jet maximum. Until recently almost nothing was known about the role these modes played in the stability spectrum of the Bickley jet and that which had been conjectured was, in fact, incorrect. Here, we investigate numerically the nonlinear evolution of “near-singular” perturbations in which the phase velocity of the initial perturbation is asymptotically near but not equal to the maximum jet velocity. We show that these modes are surprisingly stable over time. We also show that there is a clearly defined slow time oscillation in the wave number power spectrum of the perturbation stream function which is the result of a slow time oscillation in the underlying modal amplitude. For an initial near-singular mode with a nonzero phase shift across the critical levels, we show that there is a slow time oscillation in the transverse transport of perturbation energy in which the energy flux goes from one critical level to the other and then reverses and so on all the while satisfying no net energy transfer from the mean flow to the perturbation field.

Surface-wave generation: a viscoelastic model

The Reynolds-averaged equations for turbulent flow over a deep-water sinusoidal gravity wave, z = acoskx ≡ h0(x), are formulated in the wave-following coordinates ζ, η, where x = ζ, z =η + h(ζ, η), h(ζ, 0) = h0(ζ) and h is exponentially small for kη [Gt ] 1, and closed by a viscoelastic consitutive equation (a mixing-length model with relaxation). This closure is derived from Townsend's boundary-layer-evolution equation on the assumptions that: the basic velocity profile is logarithmic in η + z0, where z0 is a roughness length determined by Charnock's similarity relation; the lateral transport of turbulent energy in the perturbed flow is negligible; the dissipation length is proportional to η + z0. A counterpart of the Orr-Sommerfeld equation for the complex amplitude of the perturbation stream function is derived and used to construct a quadratic functional for the energy transfer to the wave. A corresponding Galerkin approximation that is based on independent variational approximations for outer (quasi-laminar) and inner (shear-stress) domains yields an energy-transfer parameter β that is comparable in magnitude with that of the quasi-laminar model (Miles 1957) and those calculated by Townsend (1972) and Gent & Taylor (1976) through numerical integration of the Reynolds-averaged equations. The calculated limiting values of β for very slow waves, with Charnock's relation replaced by kz0 = constant, are close to those inferred from observation but about three times the limiting values obtained through extrapolation of Townsend's results.

A boundary element approach to the 2D potential flow problem around airfoils with cusped trailing edge

The boundary integral method utilised in [1] for investigating the lifting flow past airfoils with non-vanishing trailing edge angle is adjusted herein (by imposing a different Kutta-Joukovsky condition) in order to study the incompressible inviscid flow past airfoils with cusped trailing edge.Using a linear interpolation for the normal derivative of the perturbation stream function, we reduce by discretization the integral representation of the perturbation stream function to an algebraic system. The solutions of this system are used in order to calculate the distribution of the tangential velocity and pressure coefficient over the airfoil.Comparison between calculated and analytical values of the pressure coefficient for a Joukovsky and a Von Mises profile show very good agreement.

Boundary-Layer Damping of Baroclinic Instability

Abstract A shooting method is applied to a 100-level perturbation model with an explicit Ekman layer. Growth rates for a no-slip boundary condition are similar to those found previously. It is noted that the large cross-isobaric flow of the classic Ekman solution overestimates the damping by roughly a factor of 2. For the condition of zero vertical gradient of perturbation streamfunction at the lower boundary (allowing slip), growth rates are only slightly reduced from the frictionless case. The physical bases for an implicit boundary layer, as well as the no-slip condition, are questioned.

Structures of dynamically unstable shear flows and their implications for shallow internal gravity waves

In this study, the response of a dynamically unstable shear flow with a critical level to periodic forcing is presented. An energy argument is proposed to explain the upshear tilt of updrafts associated with disturbances in two-dimensional stably stratified flows. In a dynamically unstable flow, the energy equation requires an upshear tilt of the perturbation streamfunction and vertical velocity whereU z is positive. A stability model is constructed using an iteration method. An upshear tilt of the vertical velocity and the streamfunction fields is evident in a dynamically unstable flow, which is required by energy conversion from the basic shear to the growing perturbation wave energy according to the energy argument. The momentum flux profile indicates that the basic flow is decreased (increased) above (below) the critical level. Thus, the shear instability tends to smooth the shear layer. Following the energy argument, a downshear tilt of the updraft is produced in an unstably stratified flow since the perturbation wave energy is negative. The wave energy budget indicates that the disturbance is caused by a thermal instability modified by a shear flow since the potential energy grows faster than the kinetic energy.

Nonlinear, periodic, internal waves

It is shown how to calculate large-amplitude, periodic, internal waves in a channel filled with continuously stratified fluid by using a Stokes-type amplitude expansion along the channel and a modal expansion across the channel. We obtain explicit solutions for the case where the density increases exponentially with depth. It is found that periodic waves in exponentially stratified fluid are waves of depression: as the wave increases in amplitude the wave speed decreases and the mean density of the fluid at a given height decreases. The waves are limited in height by the formation of an eddy of fluid on the upper boundary above the trough of the wave. This is consistent with the description of waves of depression given by Amick and Toland (1984). There is no backflow before the eddy forms. By contrast, Amick and Toland (1984) have shown that solitary waves in the same system are waves of elevation, limited in height by a cusp in the streamlines at some interior point. A relationship is found between the mass flux and the mass displacement. The expressions necessary to calculate the potential and kinetic energies are given. A simple analytic solution for the internal wave is presented for a fluid with a weak exponential stratification, that is when the Boussinesq approximation is appropriate. For an Nth mode wave the limiting amplitude of the perturbation streamfunction is shown to be 1/Nπ. This corresponds to a maximum streamline displacement of 2z/Nπ, where z (=0.7391) is the solution to z = cos z.

Numerical analysis of viscous shear flows around a circular cylinder.

Numerical solutions of the Navier-Stokes equations for viscous shear flows past an infinitely long circular cylinder are obtained for Reynolds numbers ranging from Re=1 to Re=40. The shear parameters, ε's investigated for each Re are 0.01, 0.05, 0.1, 0.2, 0.5 and 1. The outer boundary condition of the free stream is replaced by adding to a linear shear flow Oseen's exact solution for the perturbation stream function that has been obtained in the questions of the uniform flow, and the stream function, Ψs, on the surface is decided by imposing a condition that the pressure is single-valued. Numerical solutions for the drag, lift, moment and pressure distributions are presented, and their dependence on the shear parameter are investigated. The most interesting result is that lift gives rise to minus even in this Reynolds number range. Lastly, flow patterns are shown for Re=20.

Asymptotic Approximation and Perturbation Stream Functions in Viscous Flow Calculations

An asymptotic approximation technique is shown to substantially decrease the number of interations necessary for the accurate computation of steady solutions for viscous incompressible flows. A numerical study of flows exterior to a circular cylinder indicates that the technique reduces the required number of iterations by a factor of 10. The flux formulation of the vorticity stream function solution for the flow of a viscous incompressible fluid external to a circular cylinder requires the solution of the Poisson equation for the stream function. The use of the FACR (1), fast direct Poisson equation solver leads to serious errors in the computation of the stream function near the cylinder. However it is shown that these errors are eliminated through the introduction of a perturbation stream function.

Large-scale structures in a forced turbulent mixing layer

The large-scale structures that occur in a forced turbulent mixing layer at moderately high Reynolds numbers have been modelled by linear inviscid stability theory incorporating first-order corrections for slow spatial variations of the mean flow. The perturbation stream function for a spatially growing time-periodic travelling wave has been numerically evaluated for the measured linearly diverging mean flow. In an accompanying experiment periodic oscillations were imposed on the turbulent mixing layer by the motion of a small flap at the trailing edge of the splitter plate that separated the two uniform streams of different velocity. The results of the numerical computations are compared with experimental measurements.When the comparison between experimental data and the computational model was made on a purely local basis, agreement in both the amplitude and phase distribution across the mixing layer was excellent. Comparisons on a global scale revealed, not unexpectedly, less good accuracy in predicting the overall amplification.