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03-cnsrv

03-cnsrv - Lecture 3 Conservation Equations Applied...

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1 Lecture 3 - Conservation Equations Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org © André Bakker (2002-2006)

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2 The governing equations include the following conservation laws of physics: Conservation of mass. Newton’s second law: the change of momentum equals the sum of forces on a fluid particle. First law of thermodynamics (conservation of energy): rate of change of energy equals the sum of rate of heat addition to and work done on fluid particle. The fluid is treated as a continuum. For length scales of, say, 1 μ m and larger, the molecular structure and motions may be ignored. Governing equations
3 Lagrangian vs. Eulerian description A fluid flow field can be thought of as being comprised of a large number of finite sized fluid particles which have mass, momentum, internal energy, and other properties. Mathematical laws can then be written for each fluid particle. This is the Lagrangian description of fluid motion. Another view of fluid motion is the Eulerian description. In the Eulerian description of fluid motion, we consider how flow properties change at a fluid element that is fixed in space and time (x,y,z,t), rather than following individual fluid particles. Governing equations can be derived using each method and converted to the other form.

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4 Fluid element and properties The behavior of the fluid is described in terms of macroscopic properties: Velocity u. Pressure p. Density ρ. Temperature T. Energy E. Typically ignore (x,y,z,t) in the notation. Properties are averages of a sufficiently large number of molecules. A fluid element can be thought of as the smallest volume for which the continuum assumption is valid. x y z δ y δ x δ z (x,y,z) Fluid element for conservation laws Faces are labeled North, East, West, South, Top and Bottom 1 1 2 2 W E p p p p x p p x x x δ δ = - = + Properties at faces are expressed as first two terms of a Taylor series expansion, e.g. for p: and
5 Mass balance Rate of increase of mass in fluid element equals the net rate of flow of mass into element. Rate of increase is: The inflows (positive) and outflows (negative) are shown here: z y x t z y x t δ δ δ ρ δ δ ρδ = ) ( x y z ( ) 1 . 2 w w z x y z ρ ρ δ δ δ + ( ) 1 . 2 v v y x z y ρ ρ δ δ δ + z y x x u u δ δ δ ρ ρ - 2 1 . ) ( z x y y v v δ δ δ ρ ρ - 2 1 . ) ( y x z z w w δ δ δ ρ ρ - 2 1 . ) ( ( ) 1 . 2 u u x y z x ρ ρ δ δ δ +

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6 Continuity equation Summing all terms in the previous slide and dividing by the volume δ x δ y δ z results in: In vector notation: For incompressible fluids ∂ρ / t = 0 , and the equation becomes: div u = 0.
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03-cnsrv - Lecture 3 Conservation Equations Applied...

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