CFDa.2.equations

# CFDa.2.equations - Fluids Review TRN-1998-004 Governing...

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© Fluent Inc. 12/05/10 B1 Fluids Review TRN-1998-004 Governing Equations in Fluid Mechanics

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© Fluent Inc. 12/05/10 B2 Fluids Review TRN-1998-004 Outline Lagrangian and Eulerian descriptions of fluid motion Governing equations - Lagrangian form Additional relations Surface Forces Reynolds transport theorem Deriving the conservation of mass equation Conservation of mass, momentum, and energy equations Boundary conditions Simplifications of the Navier-Stokes equations Dimensional analysis Summary
© Fluent Inc. 12/05/10 B3 Fluids Review TRN-1998-004 Lagrangian Description of Fluid Motion 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.

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© Fluent Inc. 12/05/10 B4 Fluids Review TRN-1998-004 Eulerian 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 given point in space and time (x,y,z,t), rather than following individual fluid particles.
© Fluent Inc. 12/05/10 B5 Fluids Review TRN-1998-004 Eulerian vs Lagrangian The fundamental laws of fluid motion are easily derived in the Lagrangian framework. The solution of these equations for a specific flow is more easily carried out using the Eulerian framework, since it permits them to be expressed as a system of differential equations . We will therefore look at the fundamental equations of fluid mechanics as follows: Formulate the governing equations equations using the Lagrangian system Convert these equations from the Lagrangian system to the Eulerian system (using Reynolds Transport Theorem) - these equations can then be solved for specific flow fields of interest.

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© Fluent Inc. 12/05/10 B6 Fluids Review TRN-1998-004 Governing Equations - Lagrangian Form Consider a fixed quantity of fluid (a fluid particle), and follow this particle as it moves through space gs surroundin on fluid by done net work W fluid fer to heat trans net Q fluid on acting force body net F fluid on acting force surface net F potential) kinetic (internal energy total E momentum P mass total M volume b s = = = = + + = = = =
© Fluent Inc. 12/05/10 B7 Fluids Review TRN-1998-004 Governing Equations - Lagrangian Form (2) Conservation of Mass Mass is neither created nor destroyed in the fluid parcel Conservation of Momentum (Newton’s Second Law) The rate of change of momentum = sum of the forces acting on the fluid 0 = dt dM b s F F dt P d + =

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© Fluent Inc. 12/05/10 B8 Fluids Review TRN-1998-004 Governing Equations - Lagrangian Form (3) Conservation of Energy (First Law of Thermodynamics) The change in total energy = net heat transfer - net work done by the fluid Entropy Equation (Second Law of Thermodynamics) The change in entropy -heat transfer to system divided by temperature must be positive W Q dt dE - = 0 - T Q dt dS

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• Fall '10
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• Fluid Dynamics, Fluids Review, Fluent Inc., © Fluent Inc.

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