Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock–spherical fast/slow bubble interactions

1998 ◽  
Vol 362 ◽  
pp. 327-346 ◽  
Author(s):  
N. J. ZABUSKY ◽  
S. M. ZENG
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Mach Disk ◽  
Vortex Rings ◽  
Tvd Scheme ◽  
Downstream Side ◽  
Bubble Interactions ◽  
Vortex Layer ◽  
Mesh Resolution ◽  
Order Of Magnitude

Collapsing shock-bounded cavities in fast/slow (F/S) spherical and near-spherical configurations give rise to expelled jets and vortex rings. In this paper, we simulate with the Euler equations planar shocks interacting with an R12 axisymmetric spherical bubble. We visualize and quantify results that show evolving upstream and downstream complex wave patterns and emphasize the appearance of vortex rings. We examine how the magnitude of these structures scales with Mach number. The collapsing shock cavity within the bubble causes secondary shock refractions on the interface and an expelled weak jet at low Mach number. At higher Mach numbers (e.g. M=2.5) ‘vortical projectiles’ (VP) appear on the downstream side of the bubble. The primary VP arises from the delayed conical vortex layer generated at the Mach disk which forms as a result of the interaction of the curved incoming shock waves that collide on the downstream side of the bubble. These rings grow in a self-similar manner and their circulation is a function of the incoming shock Mach number. At M=5.0, it is of the same order of magnitude as the primary negative circulation deposited on the bubble interface. Also at M=2.5 and 5.0 a double vortex layer arises near the apex of the bubble and moves off the interface. It evolves into a VP, an asymmetric diffuse double ring, and moves radially beyond the apex of the bubble. Our simulations of the Euler equations were done with a second-order-accurate Harten–Yee-type upwind TVD scheme with an approximate Riemann Solver on mesh resolution of 803×123 with a bubble of radius 55 zones.

scholarly journals Linearly implicit all Mach number shock capturing schemes for the Euler equations

2019 ◽  
Vol 393 ◽  
pp. 278-312 ◽  
Author(s):  
Stavros Avgerinos ◽  
Florian Bernard ◽  
Angelo Iollo ◽  
Giovanni Russo
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Shock Capturing ◽  
Shock Capturing Schemes

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

A FULLY IMPLICIT SOLVER OF 3-D EULER EQUATIONS ON MULTIBLOCK CURVILINEAR GRIDS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1483-1486 ◽  
Author(s):  
HAI-QING SI ◽  
TONG-GUANG WANG ◽  
XIAO-YUN LUO
Keyword(s):  
Euler Equations ◽  
Three Dimensional ◽  
Implicit Time ◽  
Tvd Scheme ◽  
Time Step ◽  
Vector Operation ◽  
Fully Implicit ◽  
Curvilinear Grids ◽  
The Matrix ◽  
Implicit Solver

A fully implicit unfactored algorithm for three-dimensional Euler equations is developed and tested on multi-block curvilinear meshes. The convective terms are discretized using an upwind TVD scheme. The large sparse linear system generated at each implicit time step is solved by GMRES* method combined with the block incomplete lower-upper preconditioner. In order to reduce the memory requirements and the matrix-vector operation counts, an approximate method is used to derive the Jacobian matrix, which only costs half of the computational efforts of the exact Jacobian calculation. The comparison between the numerical results and the experimental data shows good agreement, which demonstrates that the implicit algorithm presented is effective and efficient.

scholarly journals A Hybrid Adaptive Low-Mach-Number/Compressible Method for the Euler Equations

2017 ◽  
Author(s):  
Emmanuel Motheau ◽  
Max Duarte ◽  
Ann S. Almgren ◽  
John B. Bell
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Low Mach Number

Numerical Simulation of Unsteady Aerodynamic Loads over an Aircraft

2014 ◽  
Vol 908 ◽  
pp. 264-268
Author(s):  
Xiao Jun Xiang ◽  
Yu Qian
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Unsteady Aerodynamics ◽  
Periodic Motions ◽  
Aerodynamic Loads ◽  
Unsteady Aerodynamic ◽  
Sinusoidal Motion ◽  
A Cell ◽  
Rigid Body Motions ◽  
Volume Method

The unsteady aerodynamic loads are the basic of the aeroelastic. This paper focuses on the computation of the unsteady aerodynamic loads for forced periodic motions under different Mach numbers. The flow is modeled using the Euler equations and an unsteady time-domain solver is used for the computation of aerodynamic loads for forced periodic motions. The Euler equations are discretized on curvilinear multi-block body conforming girds using a cell-centred finite volume method. The implicit dual-time method proposed by Jameson is used for time-accurate calculations. Rigid body motions were treated by moving the mesh rigidly in response to the applied sinusoidal motion. For an aircraft model, a validation of the unsteady aerodynamics loads is first considered. Furthermore, a study for understanding the influence of different Mach number was conducted. A reverse of the trend of hysteretic loops can be observed with the increasing of the Mach number.

A hybrid pressure-density-based algorithm for the Euler equations at all Mach number regimes

2011 ◽  
Vol 70 (8) ◽  
pp. 961-976 ◽  
Author(s):  
C. M. Xisto ◽  
J. C. Páscoa ◽  
P. J. Oliveira ◽  
D. A. Nicolini
Keyword(s):  
Mach Number ◽  
Euler Equations

scholarly journals Advective balance in pipe-formed vortex rings

2017 ◽  
Vol 836 ◽  
pp. 773-796
Author(s):  
Karim Shariff ◽  
Paul S. Krueger
Keyword(s):  
Reynolds Numbers ◽  
Vortex Rings ◽  
Quadratic Term ◽  
Axisymmetric Vortex ◽  
Order Of Magnitude ◽  
Cylindrical Diffusion ◽  
Type Boundary ◽  
The One ◽  
Type Boundary Condition ◽  
Edge Layer

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

Analysis of Steady and Unsteady Turbine Cascade Flows by a Locally Implicit Hybrid Algorithm

1992 ◽  
Author(s):  
C. J. Hwang ◽  
J. L. Liu
Keyword(s):  
Mach Number ◽  
Turbulent Flows ◽  
Hybrid Algorithm ◽  
Stokes Equations ◽  
Current Approach ◽  
Numerical Dissipation ◽  
Tvd Scheme ◽  
Blade Surface ◽  
Turbine Cascade ◽  
Flow Angle

For the two-dimensional steady and unsteady turbine cascade flows, the Euler/Navier-Stokes equations with Baldwin-Lomax turbulence model are solved in the Cartesian coordinate system. A locally implicit hybrid algorithm on the mixed type of meshes is employed, where the convective dominated part in the flowfield is studied by TVD scheme to obtain high-resolution results on the triangular elements, and the second- and fourth-order dissipative model is introduced on the O-typed quadrilateral grid in the viscous dominated region to minimize the numerical dissipation. When the steady subsonic and transonic turbulent flows are investigated, the distributions of isentropic Mach number on the blade surface, exit flow angle and loss coefficient are obtained. Comparing the present results with the experimental data, the accuracy and reliability of the current approach is confirmed. By giving a moving wake-type total pressure profile at the inlet plane in the rotor-relative frame of reference, the unsteady transonic inviscid and turbulent flows calculations are performed to understand the interaction of the upstream wake with a moving blade row. The Mach number contours, perturbation component of the unsteady velocity vectors, shear stress and pressure distributions on the blade surface are presented. The physical phenomena, which include periodical flow separation on the suction side, bowing, chopping and distortion of incoming wake, negative jet, convection of the vortices and wake segments, and vortex shedding at the trailing edge, are observed. It is concluded that the unsteady aerodynamic behaviors are strongly dependent on the wake/shock/boundary layer interactions.

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