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(RTT)

(RTT). • Most principles of fluid mechanics are adopted from solid mechanics, where the physical laws dealing with the time rates of change of extensive properties are expressed for systems. • In fluid mechanics, it is usually more convenient to work

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(RTT)

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  1. (RTT) • Most principles of fluid mechanics are adopted from solid mechanics, where the physical laws dealing with the time rates of change of extensive properties are expressed for systems. • In fluid mechanics, it is usually more convenient to work with control volumes, and thus there is a need to relate the changes in a control volume to the changes in a system. • The relationship between the time rates of change of an extensive property for a system and for a control volume is expressed by the Reynolds transport theorem (RTT), which provides the link between the system and control volume • RTT is named after the English engineer, Osborne Reynolds (1842– 1912), who did much to advance its application in fluid mechanics.

  2. The laws in their basic forms are stated in terms of system a system (or Object): a collection of matter of fixed identity (always coame atoms or molecules and always containing same mass no matter moves and interacts with its surrounding). • System approach is NOT very suitable for the applications in fluid mesh is hard to identify and track a group of fluid since moves quite instead, control volume (CV) approach is more often used in fluid mesh.

  3. How to choose a control volume? • CV is arbitrarily chosen by fluid dynamicist, however, selection of CV can either simplify or complicate analysis; • Clearly define all boundaries. Analysis is often simplified if CS is normal to flow direction; • Only CS conditions needed! Do not require detailed information inside CV. • Clearly identify all fluxes crossing the CS; • Clearly identify forces & torques of interest acting on the CV and CS.

  4. Reynolds—Transport Theorem (RTT)

  5. Reynolds—Transport Theorem (RTT) • the time rate of change of the property B of the system is equal to the time rate of change of B of the control volume plus the net flux of B out of the control volume by mass crossing the control surface.

  6. Reynolds—Transport Theorem (RTT) The total amount of property B within the control volume must be determined by integration: Therefore, the system-to-control- volume transformation for a fixed control volume:

  7. Material derivative (differential analysis): General RTT, non-fixed CV (integral analysis):

  8. Reynolds—Transport Theorem (RTT) • Interpretation of the RTT: – Time rate of change of the property B of the system is equal to (Term 1) + (Term 2) – Term 1: the time rate of change of B of the control volume – Term 2: the net flux of B out of the control volume by mass crossing the control surface

  9. Moving control volume • Fixed, moving, and deforming control volumes • For moving CV, use relative • velocity in the surface integral; • For deforming CV, use relative • velocity on all the deforming • control surfaces, • W = V −Vcv • W = V −Vcs

  10. For moving and/or deforming control volumes, • Where the absolute velocity V in the second term is replaced by the relative velocity Vr = V –VCS • Vr is the fluid velocity expressed relative to a coordinate system moving with the control volume.

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