Stability Problem For Flow Research Articles

Dynamic numerical simulation method and flow characteristic analysis on the mode transition in variable cycle compression system

Studying the dynamic aerodynamic characteristics and flow field characteristics for the dynamic mode transition in the variable cycle compression system (VCCS) can guarantee that the variable cycle engine (VCE) achieves high performance and stable mode transition. The common quasi-steady state analysis method cannot reflect the transient performance of the dynamic mode transition. Moreover, the dynamic simulation based on the dynamic grid technology will encounter the problem of negative volume grids because the adjustment range of variable components for VCCS is usually very large. To solve the problem, a dynamic simulation method based on real-time variation for geometry and aerodynamic boundaries of the dynamic mode transition in VCCS is first established in this paper. Using this method, the dynamic simulation was completed for the couple adjustment process of the outer bypass valve and variable stators of the dynamic mode transition in VCCS. The results of the dynamic simulation are compared with the quasi-steady state results. The results show that the rotor work ability and stator separation degree are affected by the stator variable angle degree and the mass flow rate (MFR) during the dynamic opening process of the outer bypass valve. For the MFR variation of 14.8%, when the stator variable angle is small (less than 3.3° in this study example), the flow separation on the variable stator blade pressure surface is increased during the dynamic opening process of the outer bypass valve. And the work ability of the downstream rotor is weakened. Therefore, in order to reduce the flow separation of stator blades with a small variable angle during the dynamic opening process of the outer bypass valve, a larger positive incidence angle can be adopted for stators at a small bypass ratio. For the whole dynamic process, the variation trend of the rotor work ability and stator separation state is opposite in the opening and closing process of the outer bypass valve. However, there is a dynamic effect of the flow field variation lagging behind the geometry variation in the opening and closing process. Therefore, the transient flow field in the opening and closing process is different from the quasi-steady state results, and the difference is more significant in the case of a small valve opening. Under a small valve opening, the work ability of the stage before the outer bypass entrance in the open process is significantly smaller than that in the closing process. And the difference in the flow field is mainly that the stator separation state of the stage before the outer bypass entrance in the open process is significantly weaker than that in the closing process. Therefore, the dynamic effect of the opening and closing process is more significant under a small valve opening, and the flow stability problem is more likely to occur during the dynamic mode transition in VCCS.

Gill's problem in a sandwiched porous slab

The classical Gill's stability problem for stationary and parallel buoyant flow in a vertical porous slab with impermeable and isothermal boundaries kept at different temperatures is reconsidered from a different perspective. A three-layer slab is studied instead of a homogeneous slab as in Gill's problem. The three layers have a symmetric configuration where the two external layers have a high thermal conductivity, while the core layer has a much lower conductivity. A simplified model is set up where the thermal conductivity ratio between the external layers and the internal core is assumed as infinite. It is shown that a flow instability in the sandwiched porous slab may arise with a sufficiently large Rayleigh number. It is also demonstrated that this instability coincides with that predicted in a previous analysis for a homogeneous porous layer with permeable boundaries, by considering the limiting case where the permeability of the external layers is much larger than that of the core layer.

Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

Derivatives of eigenvalues and eigenvectors with respect to design variables are required for gradient-based optimization in many engineering design problems. However, for the generalized and standard eigenvalue problems with general complex and non-Hermitian coefficient matrices, no method can accurately compute the eigenvalue and eigenvector derivatives while remaining efficient for large numbers of design variables. In this paper, we develop an adjoint method to compute complex eigenvalue and eigenvector derivatives with machine precision. For the special case when only the eigenvalue derivative is required, we propose a reverse algorithmic differentiation (RAD) formula using a newly developed dot product identity for complex functions. We verify the proposed method against the finite differences (FD) for a simple algebraic example with a 3-by-3 complex non-Hermitian matrix and a plane Poiseulle flow stability problem that is modeled as a generalized eigenvalue problem. The adjoint method is demonstrated to scale well with the number of design variables, matching the FD reference to about 5 to 7 digits.

LINEAR STABILITY OF A FILTRATION FLOW WITH GAS–OIL INTERFACE WITHIN THE BRINKMAN APPROACH

Abstract—The problem of vertical flow stability in an oil reservoir with a gas cap is considered, when the oil flow obeys the Brinkman equation. Boundary conditions at the moving boundary of the gas-oil interface are derived and a basic solution is obtained. The normal mode method is used to study the stability of the gas–oil interface. The obtained dispersion equation is investigated. Conditions for flow stability are found for all values of the parameters, and it is shown that, in the linear approximation, the growth rate of short-wave perturbations tends to zero with increasing wave number.

Analysis Method of Compressor Stability Based on Eigenvalue Theory

Abstract In order to analyze the flow stability of compressors rapidly in the design stage, a fast analysis method is developed in this paper. This method takes the base flow on each streamline at different spanwise locations as the research object and characterizes the effect of blade rows on the internal flow by establishing a body force model. Therefore, it is capable of taking the complex three-dimensional blade geometry and the abundant flow details into account, and what's more, it has a significantly high computational efficiency, which makes it available to engineering practice. Based on the eigenvalue theory, the flow stability problem is transformed into the eigenvalue problem, and the system stability is determined by the eigenvalue obtained by utilizing the singular value decomposition method. The flow stability of NASA Rotor 37 is analyzed to validate the reliability of the developed approach, and the results show that this analysis method is able to capture the process of stability deterioration with throttling and identify the blade tip region as the most unstable area. In order to further study the influence of blade geometry on compressor stability, a number of swept rotors are designed by modifying the stacking line of NASA Rotor 37. The assessment results via the developed method indicate that the forward sweep defined in this work can significantly improve the stability of the flow field in the tip region, while the backward sweep has the opposite effect.

Note on Instability Regions in the Circular Rayleigh Problem of Hydrodynamic Stability

Batchelor and Gill’s semicircular bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of inviscid incompressible homogeneous axial flows with respect to axisymmetric disturbances is further reduced. Significance In the circular Rayleigh problem of hydrodynamic stability, we have obtained a parabolic instability region. The significance of this result is that it depends on the discriminant $$\Psi (r)$$ which plays a vital role in deciding whether a basic axial flow is stable or not to axisymmetric disturbances. However, our parabolic instability region is an unbounded one unlike the semicircular instability region of Batchelor and Gill, which has the weakness that it does not depend on $$\Psi (r)$$ . So we have demonstrated that our parabola intersects the semicircle of Batchelor and Gill for two classes of basic flows, and consequently the semicircular instability region of Batchelor and Gill is further reduced. Moreover, we have stated the stability problem consisting of a second-order ODE and associated boundary conditions in three different but related variables. The significance of this is that further analytical results can be obtained for this problem, and some of these are presented in our subsequent paper.

Finite element analysis of aero-hydroelastic stability of arbitrary shape panels

A fluid-structure model based on Mindlin-Reissner plate theory and linearized incompressible potential flow theory is used to study the aero-hydroelastic stability of panels of arbitrary planforms. A finite-element procedure is developed to reduce the continuous system to a finite flow-structure system, which is solved in the frequency domain. The finite element method is crucial for the study of panels of arbitrary geometry and circumvents difficulties with the assumed mode shape approach. This paper is the first finite-element formulation of a fully coupled fluid-structure model of the aero-hydroelastic panel stability problem for incompressible flow. Circular, square, and triangular panels surrounded by a rigid baffle with different boundary conditions are studied. Eigenvalues of the system are surveyed as a function of the hydrodynamic loading at high and low density ratios. Critical flow velocity at divergence-onset and modal coalescence divergence or flutter are determined and compared for the three planforms. For simply supported or clamped panels of equal area and flexural rigidity, as the fluid velocity increases from zero the circular panel diverges first then the square and equilateral triangle, the later is the most stable. In order to compare the relative stability of the three geometries considered, a velocity ratio that consolidates divergence and flutter conditions for the three panels is introduced. Such a parameter offers a simple formula for the divergence velocity in terms of the panel geometry and flow and material properties, and is very useful to the designer in a wide range of engineering applications.

A spectral perspective about the accuracy of numerical predictions of flow instabilities

The paper discusses the usefulness of information provided by the eigenvalues, eigenfunctions and eigenvectors of flow stability problems, aiming to propose in a spectral perspective an evaluation of the accuracy of the related numerical solution schemes. The presented material further develops the results of past applications of numerical techniques in the analysis of the stability of heated systems containing single-phase, two-phase and supercritical pressure fluids. Since in all these applications both nonlinear and linear stability analyses were performed adopting very similar techniques, information on the different modes of oscillation or excursion exhibited by these systems during unstable behaviour was also made available. These spectral data, whose usefulness was up to now not completely exploited, are now considered in a new perspective, by addressing the simplest of the previously addressed problems both by analytical and numerical means. Information on the level of accuracy obtained by numerical solution techniques in capturing the intrinsic dynamics of the addressed flow systems is presented.

High-Gain Observer-Based Sliding-Mode Dynamic Surface Control for Particleboard Glue Mixing and Dosing System

In the process of particleboard glue mixing and dosing control under the working condition of intermediate frequency, a sliding-mode dynamic surface control strategy based on high-gain observer is proposed in this paper to deal with the problem of glue flow stability caused by strong nonlinearity. The high-gain observer (HGO) is aimed at estimating the derivative of the immeasurable system input signal for feedback, and the robustness of the system is improved by the dynamic surface control (DSC) method. Furthermore, the sliding-mode control (SMC) method is used to deal with disturbances caused by the uncertainties as well as external disturbances. It is proven that the system is exponential asymptotic stable by constructing a suitable Lyapunov function. Simulation results show that the proposed control methods can make the system track the expected flow value quickly and accurately. Finally, numerical simulation results are exhibited to authenticate and validate the effectiveness of the proposed control scheme.

On Neutral and Unstable Modes and on the Role of Reynolds Stress in the Stability Problem of Density Stratified Shear Flows in Channels with Variable Bottom

The stability problem of inviscid incompressible stratified shear flows in sea straits of arbitrary cross sections is formulated without the Boussinesq approximation. Series solutions of the stability equation are found by the method of Frobenius and these solutions are used to study the existence of neutral modes when the Richardson number is larger than one quarter. The energy aspect of the problem is studied by discussing the role of the Reynolds stress and its variation in the vertical direction for both unstable modes and neutral modes adjacent to the unstable modes. The interfacial conditions at layers of discontinuity of the basic velocity, density or topography are derived and these are used in the study of instability of two examples of basic flows.

Direct Numerical Simulation Code Validation for Compressible Shear Flows Using Linear Stability Theory

In order to simulate compressible shear flow stability and aeroacoustic problems, a numerical code must be able to capture how a baseflow behaves when submitted to small disturbances. If the disturbances are amplified, the flow is unstable. The linear stability theory (LST) provides a framework to obtain information about the growth rate in relation to the excitation frequency for a given baseflow. A linear direct numerical simulation (DNS) should capture the same growth rate as the LST, providing a severe test for the code. In the present study, DNS simulations of a two-dimensional compressible mixing layer and of a two-dimensional compressible plane jet are performed. Disturbances are introduced at the domain inflow and spatial growth rates obtained with a DNS code are compared with growth rates obtained from LST analyses, for each baseflow, in order to verify and validate the DNS code. The good comparison between DNS simulations and LST results indicates that the code is able to simulate compressible flow problems and it is possible to use it to perform numerical simulation of instability and aeroacoustic problems.

Dynamics of the two-dimensional ideal incompressible fluid and Casimirs

Problems are considered in which the role of Casimirs in forming dynamics of the two-dimensional ideal incompressible fluid is basically studied; in particular, the conditions are formulated which arise in the stability problem of two-dimensional flows in the presence of Casimirs. Some general approaches to the construction of difference schemes for solving equations of two-dimensional fluid which possess the given Casimirs are considered.

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

SummaryIn this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size.To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L2‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.

Stability of Disturbance Waves in Developing Shear Flows: A Review of ‘Ad-hoc’ Methods

This paper attempts to induce some formalism in the study of stability of developing shear flows, by use of so called ‘ad-hoc’ methods wherein all terms upto and inclusive of a particular order are lumped together in the same equation. However, whether the ‘ordering’ is perturbation or asympotic is somewhat vague. The classical example of stability of a developing shear flow is the stability problem of boundary-layer flow over a flat-plate including the non-parallel effects. Other examples are free shear flows; also, problems involving flow over alternate rigid and porous panels with suction, or suction from one wall and injection from the opposite wall. The quick method for stability analysis is to use the local Orr-Sommerfeld solution, using the quasi-parallel approximation. This is sometimes augmented by inclusion of some non-parallel terms, perhaps in an ad-hoc manner, and so called ‘improved solutions’ are obtained. The present paper attempts to establish a formal framework for these methods so that these methods are no more termed ‘ad-hoc’.

Short wave stability of homogeneous shear flows with variable topography

For the stability problem of homogeneous shear flows in sea straits of arbitrary cross section, a sufficient condition for stability is derived under the condition of inviscid flow. It is shown that there is a critical wave number, and if the wave number of a normal mode is greater than this critical wave number, the mode is stable.

On the Stability of a Compressible Axial Flow with an Axial Magnetic Field

We consider the stability problem of inviscid compressible axial flows with axial magnetic fields following the work of Dandapat and Gupta (Quarterly of Applied Mathematics, 1975). A numerical study of the stability of some basic flows has been carried out and it is found that an increase in the magnetic field strength has a stabilizing effect on subsonic flows and a destabilizing effect on supersonic flows. An analytical study of the stability problem has also been done in the present paper, but this analytical study is restricted by the approximation <svg style="vertical-align:-0.04095pt;width:49.612499px;" id="M1" height="11.225" version="1.1" viewBox="0 0 49.612499 11.225" width="49.612499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g><g transform="matrix(.017,-0,0,-.017,22.127,11.113)"><path id="x226A" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58zM780 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,41.386,11.113)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg> and <svg style="vertical-align:-3.3907pt;width:43.037498px;" id="M2" height="15.1375" version="1.1" viewBox="0 0 43.037498 15.1375" width="43.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x1D450" d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" /></g> <g transform="matrix(.012,-0,0,-.012,6.962,14.938)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30&#xA;q16 0 76 48z" /></g> <g transform="matrix(.017,-0,0,-.017,15.563,10.862)"><use xlink:href="#x226A"/></g><g transform="matrix(.017,-0,0,-.017,34.805,10.862)"><use xlink:href="#x31"/></g> </svg>, where <svg style="vertical-align:-0.0pt;width:17.475px;" id="M3" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g> </svg> is the Mach number and <svg style="vertical-align:-3.3907pt;width:10.9px;" id="M4" height="11.9625" version="1.1" viewBox="0 0 10.9 11.9625" width="10.9" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> <g transform="matrix(.012,-0,0,-.012,6.962,11.762)"><use xlink:href="#x1D456"/></g> </svg> is the imaginary part of the complex phase velocity <svg style="vertical-align:-0.1638pt;width:7.0250001px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 7.0250001 7.9499998" width="7.0250001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> </svg>. A semicircular region depending on the magnetic field parameter and the Mach number is found for subsonic disturbances and as a consequence it is found that sufficiently strong magnetic field stabilizes all subsonic disturbances. Under a weak magnetic field, it is shown that short subsonic disturbances are stable.

Recurrence formulas for long wavelength asymptotics in the problem of shear flow stability

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for two-dimensional viscous incompressible shear flows with a nonzero average. It is shown that the critical eigenvalues are odd functions of the wave number, while the critical values of the viscosity are even functions. If the deviation of the velocity from its period-average value is an odd function of spatial variable, the eigenvalues can be found exactly.

Stability of Swirl Axisymmetric Incompressible Flow

A new method of solution of the stability problem for swirl flow of a viscous incompressible liquid is developed. The method is based on the expansion of required functions into power series of radial coordinate and allows to avoid numerical integra- tion of the system of differential equations with a singular point. As an example the stability of Poiseuille flow in a rotating pipe is considered. Calculations by new method agree well with known numerical integration result.

Developing a CFD methodology for the analysis of flow stability in heated channels with fluids at supercritical pressures

The paper presents the first results of a systematic methodology aimed at assessing the feasibility of analyses by CFD codes of flow instabilities in heated channels containing supercritical fluids. The research makes use of features presently available in CFD models, in the aim to move step-by-step from simple channel cases towards the analysis of more realistic fuel bundle subchannels.In the present step, basing on previous experience, the STAR-CCM+ code is adopted to solve flow stability problems in circular channels and fuel bundle slices without heating structures, in the aim to characterise the response of CFD models in the analysis of purely thermal–hydraulic instability phenomena. Some of the effects related to numerical discretisation, flow direction with respect to gravity and fluid properties are studied, comparing the stability thresholds identified by transient calculations with maps set up by in-house 1D codes developed and adopted in previous work. Both static and dynamic instabilities are observed, clearly showing the contiguity of these two kinds of phenomena as a function of inlet fluid subcooling.Conclusions are finally drawn about the promising features of CFD codes for such applications, sketching the lines of the work already going on in order to address more realistic reactor scale conditions.

Influence of Aircraft Maneuver Preference Variability on Airspace Usage

Human factors constitute a core aspect of air traffic control concepts and must be addressed when dealing with traffic flow stability problems. This paper aims to evaluate how the usage of airspace depends on the degree of flexibility in pilot decision-making. Specifically, we study the airspace usage of two intersecting flows of aircraft when pilot maneuver preferences vary. The amount of airspace required for the two flows to cross without conflict can reveal certain aspects of traffic complexity of this specific traffic pattern. To simplify the analysis, we use models of instantaneous lateral and longitudinal position changes to approximate the heading-change and speed-change maneuvers of the aircraft, respectively. Pilot preferences, which are expressed as preferred heading or speed changes, are reflected by different penalty functions of position displacements that the pilots attempt to minimize during conflict resolution. Under the same pilot preferences, the aircraft flow stability is preserved using a decentralized conflict-resolution scheme. However, when the pilot preferences are allowed to vary so that individual aircraft have more control freedom in conflict resolution, the intersecting flows require a much larger fraction of airspace to ensure conflict-free and stable flows of aircraft.