Vortex breakdown of compressible subsonic swirling flows in a finite-length straight circular pipe

2015 ◽  
Vol 781 ◽  
pp. 3-27 ◽  
Author(s):  
Zvi Rusak ◽  
Jung J. Choi ◽  
Nicholas Bourquard ◽  
Shixiao Wang
Keyword(s):  
Differential Equation ◽  
Mach Number ◽  
Finite Length ◽  
Compressible Flows ◽  
Vortex Breakdown ◽  
Nonlinear Partial Differential Equation ◽  
Swirling Flows ◽  
Circular Pipe ◽  
Flow Profile ◽  
Inlet Flow

A global analysis of steady states of inviscid compressible subsonic swirling flows in a finite-length straight circular pipe is developed. A nonlinear partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. The equation reflects the complicated thermo–physical interactions in the flows. Several types of solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a flow force condition describe the outlet state of the flow in the pipe. These solutions are used to form the bifurcation diagram of steady compressible flows with swirl as the inlet swirl level is increased at a fixed inlet Mach number. The approach is applied to two profiles of inlet flows, solid-body rotation and the Lamb–Oseen vortex, both with a uniform axial velocity and temperature. The computed results provide for each inlet flow profile theoretical predictions of the critical swirl levels for the appearance of vortex breakdown states as a function of the inlet Mach number, suggesting that the results are robust for a variety of inlet swirling flows. The analysis sheds light on the dynamics of compressible flows with swirl and vortex breakdown, and shows the delay in the appearance of breakdown with increase of the inlet axial flow Mach number in the subsonic range of operation. The present theory is limited to axisymmetric dynamics of swirling flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

Swirling flow states of compressible single-phase supercritical fluids in a rotating finite-length straight circular pipe

2018 ◽  
Vol 849 ◽  
pp. 576-614
Author(s):  
Nguyen Ly ◽  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Differential Equation ◽  
Finite Length ◽  
Compressible Flows ◽  
Single Phase ◽  
Nonlinear Partial Differential Equation ◽  
Swirling Flows ◽  
Circular Pipe ◽  
Inlet Flow ◽  
Total Enthalpy ◽  
Inlet Swirl

Steady states of inviscid, compressible and axisymmetric swirling flows of a single-phase, inert, thermodynamically supercritical fluid in a rotating, finite-length, straight, long circular pipe are studied. The fluid thermodynamic behaviour is modelled by the van der Waals equation of state. A nonlinear partial differential equation for the solution of the flow streamfunction is derived from the fluid equations of motion in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. This equation reflects the complicated, nonlinear thermo-physical interactions in the flows, specifically when the inlet state temperature and density profiles vary around the critical thermodynamic point, flow compressibility is significant and the inlet swirl ratio is high. Several types of solutions of the resulting nonlinear ordinary differential equation for the axially independent case describe the flow outlet state when the pipe is sufficiently long. The approach is applied to an inlet flow described by a solid-body rotation with uniform profiles of the axial velocity and temperature. The solutions are used to form the bifurcation diagrams of steady compressible flows of real fluids as the inlet swirl level and the centreline inlet density are increased at a fixed inlet Mach number and temperature. Focus is on heavy-molecule fluids with low values of $R/C_{v}$. Computed results provide theoretical predictions of the critical swirl levels for the exchange of stability of the columnar state and for the appearance of non-columnar states and of vortex breakdown states as a function of inlet centreline density. The difference in the dynamical behaviour between that of a calorically perfect gas and of a real gas is explored. The analysis sheds new fundamental light on the complex dynamics of high-Reynolds-number, compressible, subsonic swirling flows of real gases.

Wall-separation and vortex-breakdown zones in a solid-body rotation flow in a rotating finite-length straight circular pipe

2014 ◽  
Vol 759 ◽  
pp. 321-359 ◽  
Author(s):  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Solid Body ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Swirling Flows ◽  
Flow Dynamics ◽  
Circular Pipe ◽  
Body Rotation ◽  
Solid Body Rotation

AbstractThe incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ${\it\omega}_{1}$. We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ${\it\omega}$. We prove that the base columnar flow is a unique steady-state solution when ${\it\omega}$ is below ${\it\omega}_{1}$. This state is asymptotically stable and a global attractor of the flow dynamics. However, when ${\it\omega}>{\it\omega}_{1}$, we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ${\it\omega}>{\it\omega}_{1}$, the base columnar flow is a min–max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min–max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007–1016), followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.

Near-critical swirling flow of a viscoelastic fluid in a circular pipe

2017 ◽  
Vol 814 ◽  
pp. 325-360
Author(s):  
Zvi Rusak ◽  
Nguyen Ly ◽  
John A. Tichy ◽  
Shixiao Wang
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Vortex Breakdown ◽  
Shear Thinning ◽  
Weissenberg Number ◽  
Swirling Flows ◽  
Circular Pipe ◽  
Viscoelastic Characteristic ◽  
Mobility Parameter ◽  
Flow Inertia

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

The supersonic and low-speed flows past circular cylinders of finite length supported at one end

1962 ◽  
Vol 12 (3) ◽  
pp. 367-387 ◽  
Author(s):  
D. M. Sykes
Keyword(s):  
Mach Number ◽  
Finite Length ◽  
Central Region ◽  
Diameter Ratio ◽  
Circular Cylinders ◽  
Form Drag ◽  
Drag Coefficients ◽  
Low Speed ◽  
Flow Past ◽  
Length To Diameter Ratio

The flow past circular cylinders of finite length, supported at one end and lying with their axes perpendicular to a uniform stream, has been investigated in a supersonic stream at Mach number 1.96 and also in a low-speed stream. In both stream it was found that the flow past the cylinders could be divided into three regions: (a) a central region, (b) that near the free end of the cylinder, and (c) that near the supported end. The locations of the second and third regions were found to be almost independent of the cylinder length-to-diameter ratio, provided that this exceeded 4, while the flow within and the extent of the first region were dependent on this ratio. Form-drag coefficients determined in the central region in the supersonic flow were in close agreement with the values determined at the same Mach number by other workers. In the low-speed flow the local form-drag coefficients were dependent on length-to-diameter ratio and were always less than that of an infinite-length cylinder at the same Reynolds number.

scholarly journals Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1268
Author(s):  
Yarong Xia ◽  
Ruoxia Yao ◽  
Xiangpeng Xin ◽  
Yan Li
Keyword(s):  
Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Dynamical Evolution ◽  
Symmetry Transformation ◽  
Nonlocal Symmetry ◽  
Similarity Reduction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Dependent Variables

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

Weakly non-linear waves in rotating fluids

1970 ◽  
Vol 42 (4) ◽  
pp. 803-822 ◽  
Author(s):  
S. Leibovich
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Vortex Breakdown ◽  
Vries Equation ◽  
Tube Wall ◽  
Rotating Fluids ◽  
Integro Differential Equation ◽  
Stationary Flows ◽  
Linear Waves ◽  
De Vries

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.

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