Steady Axisymmetric Flow Research Articles

Joint reconstruction and segmentation of noisy velocity images as an inverse Navier–Stokes problem

We formulate and solve a generalized inverse Navier–Stokes problem for the joint velocity field reconstruction and boundary segmentation of noisy flow velocity images. To regularize the problem, we use a Bayesian framework with Gaussian random fields. This allows us to estimate the uncertainties of the unknowns by approximating their posterior covariance with a quasi-Newton method. We first test the method for synthetic noisy images of two-dimensional (2-D) flows and observe that the method successfully reconstructs and segments the noisy synthetic images with a signal-to-noise ratio (SNR) of three. Then we conduct a magnetic resonance velocimetry (MRV) experiment to acquire images of an axisymmetric flow for low ( ${\simeq }6$ ) and high ( ${>}30$ ) SNRs. We show that the method is capable of reconstructing and segmenting the low SNR images, producing noiseless velocity fields and a smooth segmentation, with negligible errors compared with the high SNR images. This amounts to a reduction of the total scanning time by a factor of 27. At the same time, the method provides additional knowledge about the physics of the flow (e.g. pressure) and addresses the shortcomings of MRV (i.e. low spatial resolution and partial volume effects) that otherwise hinder the accurate estimation of wall shear stresses. Although the implementation of the method is restricted to 2-D steady planar and axisymmetric flows, the formulation applies immediately to three-dimensional (3-D) steady flows and naturally extends to 3-D periodic and unsteady flows.

Instabilities of thermocapillary flows in large Prandtl number liquid bridges between two coaxial disks with different radii

We explore the geometric effects on the thermocapillary flow instabilities in large Prandtl number (Pr = 1.4) liquid bridges between two coaxial disks with different radii under microgravity, focusing on the impacts of radius ratio Γr and aspect ratio Γ. The static deformation of the free surface is concerned by the solution of the Young–Laplace equation, and the linear stability analysis based on spectral element method is conducted for accurate identification of the instability characteristic. We observe that the flow stability is generally improved with the decrease in radius ratio Γr or aspect ratio Γ, especially for the liquid bridge heated from the upper disk. The critical oscillation frequency experiences an abrupt drop around Γr = 0.56 as Γr decreases for the liquid bridge with the bottom disk heated. Moreover, three transitions between two-dimensional axisymmetric steady flow and three-dimensional oscillatory flow are observed within the interval 0.87 < Γ ≤ 0.91 at Γr = 0.50 when the liquid bridge is heated from the upper disk. The energy analysis indicates that the instabilities for all cases are predominantly caused by the hydrothermal wave instability and the phenomenon of three transitions results from the variation of thermal energy transfer efficiency with the growth of the Marangoni number.

Effect of volume ratio on thermocapillary convection in annular liquid pools in space

In systems with liquid/liquid or liquid/gas interface under microgravity, and even in shallow liquid layers in the terrestrial conditions, thermocapillary force takes the principal role to drive natural convections. A series of numerical simulations are conducted to investigate the stability limit of axisymmetric steady thermocapillary flow in annular liquid pools with curved and adiabatic liquid surface for eight volume ratios 0.809 ≤ Vr ≤ 1.173, where Vr is defined as (liquid vol/vol of the annular gap). Simulations provide the critical temperature difference ΔTc, frequency fc, and azimuthal wave number mc for each liquid pool. At the critical condition, oscillations start in form of standing wave. At the slightly supercritical condition (ΔT*), the standing waves turn to traveling oscillations. The calculated ΔTc values decrease with the increase of Vr. A simulation code with a convective thermal boundary condition in the liquid surface suggests that heat transfer through the liquid surface significantly increases the ΔTc value.

Multiple instabilities of thermocapillary flow in a cylindrical pool with a rotating disk on the free surface

The thermocapillary flow instabilities of silicon melt in a cylindrical pool with a rotating disk on the free surface (a simplified model of the Czochralski crystal growth) are numerically investigated by using the linear stability analysis. The complete neutral or critical stability curves are determined. Results show that the neutral stability curves form a closed region in the parameter plane, in which the steady axisymmetric flow is linearly stable. Two types of rotating wave (RW1 and RW2) instabilities and two types of hydrothermal wave (HTW1 and HTW2) instabilities are found. The energy analysis shows that all the instabilities are hydrodynamic (inertial) in nature. Specifically, RW1 and RW2 are caused by the azimuthal shear induced by the crystal rotation, while HTW1 and HTW2 are caused by the radial shear induced by the thermocapillary force.

Energy transfer mechanism during ethanol evaporation with external tangential temperature gradient under low pressure environment

For the purpose of revealing the energy-transfer process during ethanol evaporation under low pressure environment, experimental measurement and numerical simulation are adopted for ethanol evaporation at different liquid depths and external radial temperature differences in annular pools. The results show that two thermocapillary vortexes appear below the liquid–vapor interface. Most of the latent heat of vaporization is from the liquid phase, but little from the gas phase. With the increase of liquid layer depth, the steady axisymmetric flow will transform into three-dimensional time-dependent convection. The mechanism of flow instability should be attributed to the variation lag of velocity behind the variation of flow resistance. The critical vapor pressure of flow destabilization depends on radial temperature difference and liquid layer depth. Furthermore, the magnitude of temperature discontinuity across the liquid–vapor interface is the largest near the hot inner cylinder. With the increase of the liquid layer depth, the temperature discontinuity decreases.

Examples of Steady Axisymmetric Flows of an Ideal Incompressible Fluid

Examples of Steady Axisymmetric Flows of an Ideal Incompressible Fluid

DRAG ON A POROUS SPHERE EMBEDDED IN COUPLE STRESS FLUID

In this paper, the flow developed by the steady axisymmetric Stokes flow of couple stress fluid through a porous sphere is analyzed. Expressions for the velocity and couple stress tensor components are gotten as far as Macdonald functions and Gagenbauer’s polynomials. An analytical solution for the flow field as far as stream function is explored. The drag force experienced by a porous sphere due to the internal and external flow is determined. The impact on drag force for various parameters like couple stress parameter and viscosity are depicted graphically. Some special cases were likewise discussed.

Homann Flow and Heat Transfer Through a Porous Medium with Uniform Suction or Injection

The steady axisymmetric flow through a porous medium of an incompressible viscous fluid impinging on a porous flat plate is examined with heat transfer. A uniform suction or injection is applied perpendicular to the plate which is kept at a constant temperature. The governing nonlinear momentum and energy equations are solved numerically using the method of finite difference. The effect of the porosity of the medium and the uniform suction or injection on both the flow and heat transfer is investigated.

Numerical investigation of atmospherelike flows in a spherical geometry.

Buoyancy-driven convection flows play a crucial role in global heat and momentum transport in atmosphere. Simplified planetary and stellar atmospheres can be described by a spherical gap geometry with special boundary conditions for the temperature. In the spherical gap, the dielectrophoretic effect is used to synthesize the radial gravity field. Lateral thermal boundary conditions are used to model solar radiation at the equator and at the poles. The temperature reaches a maximum value at the equator and becomes colder near the poles. In the case of a rotating gap, the influence of the Coriolis and centrifugal forces are taken into account. Different regimes of the two-dimensional steady basic flow are discussed in dependence on the Taylor number and Rayleigh number and for the radii ratio η=R_{in}/R_{out}, where R_{in},R_{out} are the radii of the inner and outer surfaces, respectively. Linear instability theory is used to study when the basic flow becomes unstable. The critical Rayleigh number at which the steady axisymmetric basic flow becomes time-dependently axisymmetric or three dimensional is found to be a function of the Taylor number. Furthermore, the critical azimuthal wave number m_{c}, which is responsible for the structure of the supercritical three-dimensional flow, and the critical frequency of the perturbation ω_{c} were found. The spatial location of the perturbation helps to understand the origin of the instability.

Correction to: Modelling of Mach reflections in internal axisymmetric steady supersonic flow

Correction to: Modelling of Mach reflections in internal axisymmetric steady supersonic flow

Creeping flow of couple stress fluid past a fluid sphere with a solid core

AbstractThe steady axisymmetric uniform flow of an incompressible couple stress fluid past a fluid sphere with a solid core has been studied analytically. No‐slip boundary condition and vanishing couple stress condition are applied on the surface of the sphere. The problem is formulated using the V. K. Stokes equation describing the flow outside the fluid sphere as well as the flow inside the fluid sphere. The stream function solution for the axisymmetric motion and pressure distribution for both regions are solved explicitly. The stresses acting on the fluid sphere are estimated and the drag force experienced by a fluid sphere is determined. Numerical study is undertaken to study the effect of couple stress parameter and viscosity coefficients on the drag and the results are presented in graphical forms. The results reveal that drag force, pressure and velocity of the fluid decreases in the presence of couple stresses. Some special cases of flow past a fluid sphere have been validated with earlier well‐known cases.

Boundary-layer viscous correction method for hypersonic forebody/inlet integration

The aerodynamic shape of the hypersonic forebody/inlet integrated vehicle is usually designed according to the inviscid flow. However, due to the effect of air viscosity, there is a boundary layer flow near the wall surface. In order to get the actual flow characteristics and performance parameters of the inviscid design vehicle close to the inviscid design values, the inviscid design surface must be corrected by considering the viscous boundary layer. In this paper, the theory and method of the boundary-layer viscous correction (BLVC) for the forebody/inlet integrated configuration are studied. Firstly, two modification schemes of the BLVC for the forebody/inlet integrated configuration are proposed. The calculation method of boundary-layer displacement thickness (BLDT) based on the momentum integral equation of steady two-dimensional (2D) axisymmetric compressible flow is described in detail, and two calculation methods of the BLDT are compared and analyzed. Secondly, the numerical simulation method used in this paper and its validation are introduced. Thirdly, an axisymmetric configuration of the forebody/inlet integration is selected as the first case, and two modification schemes of BLVC are studied and analyzed by numerical simulation. The effects of the two modification schemes of BLVC on the internal and external flow characteristics and aerodynamic performance of the axisymmetric configuration of the forebody/inlet integration are compared and analyzed. Finally, a full-waverider configuration of the forebody/inlet integration is selected as the second case, and the second modification scheme of the BLVC is used to modify the full-waverider configuration. The shock wave characteristics and inlet performance parameters of the modified full-waverider configuration are studied by both the numerical simulation and the wind tunnel experiment. The obtained results prove that the calculation method of the BLDT by a numerical integration method to solve the momentum integral equation of the steady 2D axisymmetric compressible flow is correct and the BLVC method and scheme for the forebody/inlet integrated configuration are effective. The modification scheme 1 of the BLVC can only improve the characteristics of forebody shock wave incident on the inlet lip under viscous conditions, that is, to improve the performance of inlet to capture precompressed air flow, while the modification scheme 2 of the BLVC can simultaneously improve the characteristics of forebody shock wave incident on the inlet lip and the characteristics of the lip shock wave cancelled on the shoulder of the inlet under viscous conditions, and make the air flow capture performance of the inlet and the pressure rise ratio at the inlet exit close to those of the inviscid design state.

Instabilities of thermocapillary-buoyancy flow in a rotating annular pool for medium-Prandtl-number fluid.

The instabilities of the steady axisymmetric thermocapillary-buoyancy flow in a rotating annular pool were investigated by linear stability analysis. The critical instability parameters for the thermocapillary-buoyancy flow (normal gravity) and the pure thermocapillary flow (microgravity) were compared under different pool depths and rotation rates. The results show that the thermocapillary-buoyancy flow is more stable than the pure thermocapillary flow due to the stabilizing effect of the gravity (buoyancy) force. Two types of oscillatory instabilities were observed depending on the different rotation rates, and the propagation direction of the hydrothermal wave is also affected by the rotation rate.

Numerical investigation on the flow around an inclined prolate spheroid

Numerical simulations are performed for the flow around an inclined 5:2 prolate spheroid in a uniform freestream. The Reynolds number (Re = 300, 500, 700, and 1000) and incidence angle (α = 0°–90°) are considered as significant parameters affecting the wake transitions, where α = 0° indicates flow parallel to the major axis of the prolate spheroid, and the Re is based on the inflow velocity U0 and the volume-equivalent sphere diameter De of the spheroid. In the range considered of Re and α, eight flow regimes are identified: (i) steady axisymmetric (SA) flow regime; (ii) steady planar symmetric flow regime; (iii) steady asymmetric (SAS) flow regime; (iv) periodic planar symmetric flow regime with non-zero mean lift or “Zig-zig-like” (Zz-like) mode; (v) periodic asymmetric flow regime with double-sided vortex shedding; (vi) multi-periodic asymmetric flow regime with double-sided vortex shedding and low frequency modulation (MPADL); (vii) multi-periodic asymmetric flow regime with single-sided vortex shedding and low frequency modulation (MPASL); and (viii) weakly chaotic state. Three of them are new and first reported, i.e., SAS, MPADL, and MPASL modes. The wake structure of the Zz-like mode is different from that of the zig-zig mode in the sphere/disk wake with a pair of streamwise vortices extending to the near wake. It is found that the elongated body can delay the onset of unsteadiness at small incidence angles. A flow regime map in the considered (Re, α) space is then provided. Finally, the physical mechanisms of the low-frequency phenomena observed at different wake modes are explored.

Some general solutions for linear Bragg–Hawthorne equation

Linear cases of Bragg–Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well-known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.

Benchmark for scale-resolving simulation with curved walls: the Taylor Couette flow

Flow between rotating concentric cylinders, or the Taylor Couette flow, has been studied extensively because of its rich physics, ranging from axisymmetric steady laminar flow, to fully developed turbulent flow. In the present study, we advocate the use of this problem as a benchmark case for scale-resolving simulation, such as large eddy simulation (LES) and direct numerical simulation (DNS). The problem is attractive because of its simple geometry, simple boundary conditions, and complex physics involving wall-shear induced and centrifugal instability. Unlike the well-known fully developed channel flow, this problem has a curved wall boundary, and it is unnecessary to add a source term to the governing equations to sustain the fully developed turbulent flow. A p-refinement study for Re = 4000 is performed first to establish DNS data, including the time history of enstrophy, which can be used as an accuracy and resolution indicator to evaluate numerical methods, and is orders of magnitude faster than using the mean flow quantities and Reynolds stresses to evaluate solution quality. Finally, an hp-refinement study is performed to establish the relative accuracy and efficiency of high-order schemes of various accuracy.

On the Origin of Geostrophic Liquid Cylinders in Rotating Sphere with Oscillating Inner Core

The effect of non-axisymmetric inertial waves on the zonal flow structure in a rotating spherical shell is studied experimentally. The wave source is the inner core, which performs circular oscillations in the equatorial plane. The case of positive frequencies that corresponds to the advanced core motion is considered. As a result of non-linear effects, a quasi-two-dimensional axisymmetric steady flow with a complex distribution of angular velocity is generated. The extrema in the velocity profile are equivalent to the nested liquid geostrophic cylinders, which appearance is due to the interaction of the inertial waves within a viscous boundary layer. One of the extrema is associated with the critical latitude at the core boundary, where the inertial waves are excited. With an increase in the oscillation frequency, the position of this extremum gradually shifts to the rotation axis. Additional geostrophic circulation occurs close to the points where the inertial wave reflects from the cavity boundary.

Counterflow slip in a two-fluid whirlpool

This experimental study reveals a counterflow slip at the interface of two immiscible fluids filling a sealed vertical cylindrical container whose bottom disk rotates while other walls are stationary. An advanced particle image velocimetry helps us analyze and quantify streamline patterns and velocity profiles of the steady axisymmetric flows of both fluids. As the angular velocity of a disk increases, starting from zero, three changes occur in topology of the upper-fluid flow while topology of the lower-fluid flow remains invariant. These changes result in the counterflow slip at the entire interface, whose nature is speculatively discussed and requires further research.

Linear stability analysis of Rayleigh–Bénard convection for cold water near its density maximum in a cylindrical container

In this paper, the onset of Rayleigh–Bénard convection of cold water near its density maximum in a vertical cylindrical container and the stability of the steady axisymmetric flow were studied by using linear stability analysis. The results show that the critical Rayleigh number for the onset of convection increases with increasing the density inversion parameter Tm (Tm=(Фmax-Фc)/(Фh-Фc), where Фmax is the temperature at the maximum density, and Фh and Фc are the temperature at the bottom wall and top wall, respectively), and some new flow patterns not found in the Rayleigh–Bénard convection of common fluids are presented.Within a certain range of density inversion parameters, the onset of convection is steady and axisymmetric. The steady axisymmetric basic flow is found to depend on the initial condition, and two distinctly different basic flow are obtained at identical parameters, due to the top-bottom symmetry breaking. For the upward basic flow, four different flow patterns are observed after the axisymmetry-breaking instability. In addition, a hysteresis phenomenon of the flow pattern transition is found when Tm=0.57. For the downward basic flow, six flow patterns are observed after the axisymmetry-breaking instability. In particular, it is found that there still exists stable axisymmetric flow beyond the second bifurcation in certain ranges of Rayleigh numbers. The energy analysis shows that both the primary instability and the axisymmetry-breaking instability are caused by the thermal-buoyancy mechanism.

KELLER-BOX SOLUTION OF THE STAGNATION POINT MICROPOLAR FLUID FLOW BETWEEN POROUS PLATES WITH INJECTION

The present study deals with the steady axisymmetric flow of micropolar fluid between two parallel porous plates when the fluid is injected through both walls at the same rate. The influence of velocity slip at the porous surface is analyzed. A detailed finite-difference solution is developed for the resulting non-linear coupled differential equations representing velocities and microrotation. The numerical computations are obtained for radial, axial velocities, and microrotation for varying injection Reynolds number, micropolar parameter, and slip coefficient. Further, a comparison of the results is given with those obtained in the literature with different methods as special cases.