Transpiration Boundary Conditions for a Steady Inverse Method

2012 ◽  
Author(s):  
Jinguang Yang ◽  
Hu Wu
Keyword(s):  
Boundary Conditions ◽  
Inverse Method ◽  
Solid Wall ◽  
Design Tool ◽  
Navier Stokes ◽  
Separation Flow ◽  
Time Cost ◽  
Navier Stokes Equation ◽  
Tangency Condition ◽  
Turbomachinery Blades

Computational fluid dynamics has been widely used in the analysis of turbomachinery blades, however, its use as a design tool is far from sophisticated. The inverse method is such a design approach, which lends it self to the latter category. One application of the inverse method is the so called “pure inverse methd”, which differs from common analysis solver mainly in the boundary conditions on the blade surfaces. For this application, the usual non-penetration boundary conditions on the blade surfaces are aborted, instead, some aerodynamic constraints are imposed, and the flow is allowed to transpire through the actual solid wall. A camber line generation equation is added to periodically re-generate the blade camber line and drive the normal velocities on the blade surfaces to zero. When converged, the inverse method should obtain the blade shapes which satisfy the specified aerodynamic performance. In the present paper, three transpiration boundary conditions for turbomachinery blades design are compared in terms of time cost, robustness, capability of coping separation flow etc. The first inverse boundary condition is based on the flow-tangency condition on the blade surfaces, the second relys on the propagating characteristics in the flow field, and the third is a hybrid version of the first and the second. The computation is validated for 2D Navier-Stokes equation. Two compressor cascades are taken as examples to compare the performance of the three transpiration boundary conditions. Finally some conclusions are drawn.

scholarly journals The Seepage Model Considering Liquid/Solid Interaction in Confined Nanoscale Pores

Geofluids ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Xiaona Cui ◽  
Erlong Yang ◽  
Kaoping Song ◽  
Yuming Wang
Keyword(s):  
Solid Wall ◽  
Pore Wall ◽  
Interaction Coefficient ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Confined Space ◽  
Navier Stokes Equation ◽  
Capillary Radius ◽  
Seepage Model ◽  
Shale Reservoirs

Different from conventional reservoirs, nanoscale pores and fractures are dominant in tight or shale reservoirs. The flow behaviors of hydrocarbons in nanopores (called “confined space”) are more complex than that of bulk spaces. The interaction between liquid hydrocarbons and solid pore wall cannot be neglected. The viscosity formula which is varied with the pore diameter and interaction coefficient of liquids and solids in confined nanopores has been introduced in this paper to describe the interaction effects of hydrocarbons and pore walls. Based on the Navier-Stokes equation, the governing equation considered liquid/solid effect in two dimensions has been established, and approximate theoretical solutions to the governing equations have been achieved after mathematic simplification. By introducing the vortex equation, the complex numerical seepage model has been discretized and solved. Numerical results show that the radial velocity distribution near the solid wall has an obvious change when considering the liquid/solid interaction. The results consist well with that approximate mathematical solution. And when the capillary radius is smaller, the liquid and solid interaction coefficient n is greater. The liquid and solid interaction obviously cannot be neglected in the seepage model if the capillary radius is small than 50 nm when n>0.1. The numerical model has also been further validated by two types of nanopore flow tests: from pore to throat and inversely from throat to pore. There is no big difference in flow regularity of throat to pore model considering when liquid/solid interaction or not, whereas the liquid/solid interaction of pore to throat model totally cannot be overlooked.

Large-frequency global regularity for the incompressible Navier–Stokes equation

1999 ◽  
Vol 129 (5) ◽  
pp. 903-912
Author(s):  
Joel D. Avrin
Keyword(s):  
Boundary Conditions ◽  
Global Regularity ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Thin Domains ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Incompressible Navier Stokes Equation

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

Numerical Investigation of a Transient Operation of a Heat Pipe With Experimental Validation

2004 ◽  
Author(s):  
Gustavo Gutierrez ◽  
Juan Catan˜o ◽  
Tien-Chien Jen
Keyword(s):  
Heat Flux ◽  
Heat Pipe ◽  
Stokes Equations ◽  
Solid Wall ◽  
Control Volume ◽  
Navier Stokes ◽  
Interface Temperature ◽  
Vapor Flow ◽  
Navier Stokes Equation ◽  
Wick Structure

In this paper, a full transient analysis of the performance of a heat pipe with a wick structure is performed. For the vapor flow, the conventional Navier-Stokes equations are used. For the liquid flow in the wick structure, which is modeled as a porous media, volume averaged Navier-Stokes equation are adopted. The energy equation is solved for the solid wall and wick structure of the heat pipe. The energy and momentum equations are coupled through the heat flux at the liquid-vapor interface that defines the suction and blowing velocities for the liquid and vapor flow. The evolution of the vapor-liquid interface temperature is coupled through the heat flux at this interface that defines the mass flux to the vapor and the new saturation conditions to maintain a fully saturation vapor all the time. A control volume approach is used in the development of the numerical scheme. A parametric study is conducted to study the effect of different parameters that affect the thermal performance of the heat pipe. And experimental setup is developed and numerical results are validated with experimental data. The results of this study will be useful for the heat pipe design and implementation in processes that are essentially transient and steady state conditions are not reached like for example drilling applications.

scholarly journals On a 2D stochastic Euler equation of transport type: Existence and geometric formulation

2014 ◽  
Vol 15 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ana Bela Cruzeiro ◽  
Iván Torrecilla
Keyword(s):  
Boundary Conditions ◽  
Euler Equation ◽  
Multiplicative Noise ◽  
Dirichlet Boundary ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Geometric Formulation ◽  
Transport Type

We prove weak existence of Euler equation (or Navier–Stokes equation) perturbed by a multiplicative noise on bounded domains of ℝ2 with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are H1 regular. The equations are of transport type.

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

1957 ◽  
Vol 2 (3) ◽  
pp. 237-262 ◽  
Author(s):  
Ian Proudman ◽  
J. R. A. Pearson
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Small Reynolds Numbers ◽  
Cylindrical Polar Coordinates ◽  
Flow Past A Sphere

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

On the Navier-Stokes equation with boundary conditions based on vorticity

2004 ◽  
Vol 269-270 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Hamid Bellout ◽  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Boundary Conditions ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation

scholarly journals Mathematical problems in modeling artificial heart

1995 ◽  
Vol 1 (3) ◽  
pp. 245-254 ◽  
Author(s):  
N. U. Ahmed
Keyword(s):  
Blood Flow ◽  
Boundary Conditions ◽  
Stokes Equation ◽  
Artificial Heart ◽  
Moving Boundary ◽  
Hyperbolic Type ◽  
The Other ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Heart Chamber

In this paper we discuss some problems arising in mathematical modeling of artificial hearts. The hydrodynamics of blood flow in an artificial heart chamber is governed by the Navier-Stokes equation, coupled with an equation of hyperbolic type subject to moving boundary conditions. The flow is induced by the motion of a diaphragm (membrane) inside the heart chamber attached to a part of the boundary and driven by a compressor (pusher plate). On one side of the diaphragm is the blood and on the other side is the compressor fluid. For a complete mathematical model it is necessary to write the equation of motion of the diaphragm and all the dynamic couplings that exist between its position, velocity and the blood flow in the heart chamber. This gives rise to a system of coupled nonlinear partial differential equations; the Navier-Stokes equation being of parabolic type and the equation for the membrane being of hyperbolic type. The system is completed by introducing all the necessary static and dynamic boundary conditions. The ultimate objective is to control the flow pattern so as to minimize hemolysis (damage to red blood cells) by optimal choice of geometry, and by optimal control of the membrane for a given geometry. The other clinical problems, such as compatibility of the material used in the construction of the heart chamber, and the membrane, are not considered in this paper. Also the dynamics of the valve is not considered here, though it is also an important element in the overall design of an artificial heart. We hope to model the valve dynamics in later paper.

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