Vibroacoustic Simulation. Alexander Peiffer

(2.1)

      for mass conservation. Expanding the second term on the right hand side in a Taylor series gives

      and finally

(2.2)

      This one dimensional equation of mass conservation in x-direction can be extended to three dimensions:

(2.3)

      The second term of (2.3) may be confusing, but it says that the change of density is not only determined by a gradient in the velocity field but also by a gradient of the density.

      2.2.2 Newton’s law – Conservation of Momentum

The same procedure is applied to the momentum of the fluid. As shown in Figure 2.2 we get for flow in x-direction:

      1 The momentum of the control volume is ρvxdV=ρvxAdx.

      2 The momentum flow into the volume (ρvx2A)x.

      3 mass flow out of the volume (ρvx2A)x+dx.

      4 The force at position x is (PA)x.

      5 The force at position x+dx is (PA)x+dx.

      6 External volume force density fx.

Figure 2.2 Momentum flow in x-direction through control volume. Source: Alexander Peiffer.

      Thus, the conservation of momentum in x reads

       (2.4)

      Using Taylor expansions for (ρvx2A)x+dx and (PA)x+dx gives

       (2.5)

      Here, fx=Fx/(Adx) is the volume force density (force per volume). Using the chain law the partial derivatives of the first and second term lead to

(2.6)

      The term in brackets is the homogeneous continuity Equation (2.1), and Equation (2.6) simplifies to

       (2.7)

      As with the conservation of mass, this can be extended to three dimensions:

(2.8)

      This equation is the non-linear, inviscid momentum equation called the Euler equation.

      2.2.3 Equation of State

      The above equations relate pressure, velocity and density. For further reducing this set we need a third equation. The easiest way would be to introduce the . Here we start with the first law of thermodynamics in order to show the difference between isotropic (or adiabatic) equation of state and other relationships.

(2.9)

With the following specific quantities per unit mass

       (2.10)

      With the specific entropy ds=dq+drT we get:

      

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