Ghosh - 550Page 111/14/2018

Euler’s Equation

The Navier-Stokes equations that were derived for incompressible flows may be applied for any Newtonian fluid whether it is viscous or inviscid. Since real fluids are viscous, it may seem useless to study inviscid flows. However, if you wish to understand the real behavior of fluid flows, the study of inviscid flows becomes extremely important. Also, there are certain viscous flows, which behave like an inviscid flow for all practical purposes. For example, flows outside the viscous boundary layer fall into this category. Even today, most of the aerodynamic analysis is based upon ideal flow theory. Therefore, we will conduct a formal study of the procedures in ideal flow analysis.

Let us begin by simplifying the Navier-Stokes equations by neglecting the viscous term. The resulting equations, called the Euler’s equations are given in x, y, and z-components as:

x:

y:

z:

Note that the above equations are unsolvable (since there are 4 unknowns- u, v, w, and p) till we add one more equation to the set. This is the incompressible continuity equation:

We can solve this set of 4 equations in 4 unknowns by computational fluid dynamics approaches. However, the procedures for hand calculations will be quite cumbersome. Instead, we will adopt some special techniques to solve this set in this chapter on ideal flows. For now, let us realize that if we are given the velocity field,Euler’s equations can be an important means to obtain the pressure field in fluid flows.

Let us illustrate the use of Euler's equations in an example.

Example (Use of the Euler's equations):

In a frictionless, incompressible flow, the velocity field in m/s and the body force are given by and ; the coordinates are measured in meters. The pressure is at point . Obtain an expression for the pressure field, .

  1. Statement of the Problem

a)Given

  • Frictionless, incompressible flow
  • Velocity field:
  • Body force:
  • The pressure: at point

b)Find

  • Obtain an expression for the pressure field, .
  1. System Diagram

It is not necessary for this problem.

  1. Assumptions
  • Steady state condition
  • Incompressible fluid flow
  • Inviscid (frictionless) fluid flow
  1. Governing Equations
  • Euler's equation:

In components:

x components:

y components:

z components:

  1. Detailed Solution

Velocity field, , shows the components to be:

Body force, , shows the components to be:

Euler's equation can be re-written as:

x component:

y component:

z component:

From calculus, it can be said that .

Using the results obtained above, the pressure dp can be written as:

Since at point ,

Finally,

  1. Critical Assessment

Remember that the pressure is a scalar quantity (not a vector quantity with components). But it varies with all three coordinates. Thus it is 3-dimensional.

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