Cartesian Navier-Stokes Equations

3 CARTESIAN NAVIER-STOKES EQUATIONS B.3.1 Mass Continuity 

---

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

Equation A.22 is the Navier-Stokes equation for the x-component of motion in rectangular Cartesian coordinates. The corresponding equations for they and z components are obvious. 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

These are called the incompressible Navier-Stokes equations in Cartesian coordinates. 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

Notice that in these equations the terms on the right-hand side are written with a certain resemblance to the rows of the stress tensor, Eq. 2.180. The pressure gradients have been written as a separate terms. In the r and 9 equations, the final term collects some of the left-overs in going from Eqs. 3.57 and 3.58, yet maintaining the other terms in a form analogous to the stress tensor. The z equation has no left-over terms, which is also the case for the Navier-Stokes equations in cartesian coordinates. 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

In this theory, equilibrium flow is obtained using thin shear layer (TSL) approximation of the governing Navier- Stokes equation. However, to investigate the stability of the fluid dynamical system the disturbance equations are obtained from the full time dependent Navier- Stokes equations, with the equilibrium condition defined by the steady laminar flow. We obtain these in Cartesian coordinate system given by. 

A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity in cartesian coordinates... 

Equation (3-10), which we have derived from the Navier-Stokes equations, governs the unknown scalar velocity function for all unidirectional flows, i.e., for any flow of the form (3-1). However, instead of Cartesian coordinates (x, y, z), it is evident that we could have derived (3-10) by using any cylindrical coordinate system (q, 1/2, z) with the direction of motion coincident with the axial coordinate z. In this case,... 

Let us consider a laminar steady-state fluid flow in a rectilinear tube of constant cross-section. The fluid streamlines in such systems are strictly parallel (we neglect the influence of the tube endpoints on the flow). We shall use the Cartesian coordinates X, Y, Z with Z-axis directed along the flow. Let us take into account the fact that the transverse velocity components of the fluid are zero and the longitudinal component depends only on the transverse coordinates. In this case, the continuity equation (1.1.1) and the first two Navier-Stokes equations in (1.1.2) are satisfied automatically, and it follows from the third equation in (1.1.2) that... 

The Navier-Stokes equations in Cartesian coordinates have the form of Eqs. (1.1.2). These are considered in conjunction with the continuity equation (1.1.1). 

These are the general Navier-Stokes equations in Cartesian coordinates. There are only three equations for the four variables p, u, v, and w, but a fourth relation is supplied by the continuity equation ... 

By replacing u with U + u etc. in the Navier-Stokes equation and taking time average, it can be shown that for the turbulent case the two-dimensional Navier-Stokes equation in cartesian coordinates becomes... 

The key of the simulation is to solve the continuity and navier-stokes equations in an Eulerian Cartesian coordinate system. Driving forces from the fluid flow are applied to the particles as body forces. These forces are also added to the fluid equations and cause change in momentum, as reflected by the change in the pressure gradient in the flow direction. 

By combining Eqs. (B.2)-(B.5), one obtains the Navier-Stokes equations (named after Claude Louis Marie Henri Navier und George Gabriel Stokes) in their Cartesian notation ... 

Equations 6.30 through 6.32 represent the complete momentum or the Navier-Stokes equations written in the conservation form of the Cartesian coordinate system for time-dependent, compressible, and viscous flow. 

If we consider the flow with constant density, we can cancel items from the continuity equation and effect of terms due to constant property and neglect body forces. Thus, we can derive the momentum or Navier-Stokes equations written in the conservation form of the Cartesian coordinate system for time-dependent, incompressible, and viscous flow in terms of velocities. The x component of the momentum equations is... 

To analyze the flow around the bubble, the Navier-Stokes equation must be solved. The Navier-Stokes equation in the Cartesian coordinates may be written as follows ... 

Expanded in fuU, the Navier-Stokes equations are three simultaneous, nonlinear scalar equations, one for each component of the velocity field. In Cartesian coordinates. Equation 7.3 takes the form... 

For a Newtonian fluid, the Navier-Stokes equations are derived by introducing the constitutive equation [1.7] into the fundamental law of mechanics. The Navier-Stokes equations in a Cartesian coordinate system are also shown in Table 1.1. 

---