Vibroacoustic Simulation - Alexander Peiffer

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rho v Subscript normal x Baseline upper A right-parenthesis Subscript x plus d x Baseline plus ModifyingAbove m With dot"/> (2.1)

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

StartFraction partial-differential left-parenthesis rho upper A d x right-parenthesis Over partial-differential t EndFraction equals left-bracket left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline minus left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline minus StartFraction partial-differential left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline Over partial-differential x EndFraction d x right-bracket plus ModifyingAbove m With dot

      and finally

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

      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

      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.

      Thus, the conservation of momentum in x reads

       StartFraction partial-differential left-parenthesis rho v Subscript normal x Baseline upper A d x right-parenthesis Over partial-differential t EndFraction equals left-parenthesis rho v Subscript normal x Superscript 2 Baseline upper A right-parenthesis Subscript x Baseline minus left-parenthesis rho v Subscript normal x Superscript 2 Baseline upper A right-parenthesis Subscript x plus d x Baseline plus left-parenthesis upper P upper A right-parenthesis Subscript x Baseline minus left-parenthesis upper P upper A right-parenthesis Subscript x plus d x Baseline plus upper F Subscript x (2.4)

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

       StartFraction partial-differential left-parenthesis rho v Subscript normal x Baseline right-parenthesis Over partial-differential t EndFraction equals minus StartFraction partial-differential left-parenthesis rho u Subscript x Superscript 2 Baseline right-parenthesis Over partial-differential x EndFraction minus StartFraction partial-differential upper P Over partial-differential x EndFraction plus f Subscript x (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

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

       rho StartFraction partial-differential v Subscript normal x Baseline Over partial-differential t EndFraction plus rho v Subscript normal x Baseline StartFraction partial-differential v Subscript normal x Baseline Over partial-differential x EndFraction plus StartFraction partial-differential upper P Over partial-differential x EndFraction equals f Subscript x (2.7)

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

      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.

       StartLayout 1st Row 1st Column d q 2nd Column period period period 3rd Column specific heat q equals q left-parenthesis upper T comma rho right-parenthesis 2nd Row 1st Column d v 2nd Column period period period 3rd Column specific volume v equals upper V slash upper M 3rd Row 1st Column upper P d v 2nd Column period period period 3rd Column specific expansion work 4th Row 1st Column d r 2nd Column period period period 3rd Column specific friction losses EndLayout (2.10)

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

       StartLayout 1st Row 1st Column d s equals left-parenthesis StartFraction partial-differential u Over partial-differential upper T EndFraction right-parenthesis Subscript upper T Baseline StartFraction d upper T Over upper T EndFraction 2nd Column plus 3rd Column StartFraction upper P Over upper T EndFraction d upper V 2nd Row 1st Column equals StartFraction c Subscript normal v Baseline Over upper T EndFraction d upper T 2nd Column plus 3rd Column StartFraction upper P Over upper T EndFraction d upper V 3rd Row 1st Column equals StartFraction c Subscript <p style= Скачать книгу