Equations of Fluid Dynamics - Computational Fluid Dynamics - Lecture Slides, Slides of Dynamics

Download Equations of Fluid Dynamics - Computational Fluid Dynamics - Lecture Slides and more Slides Dynamics in PDF only on Docsity! This image cannot currently be displayed. Equations of Fluid Dynamics, Physical Meaning of the terms, Forms suitable for CFD Equations are based on the following physical principles: .Mass is conserved .Newton’s Second Law: F = ma .The First Law of thermodynamics: ∆e = δq - δw, for a system. Docsity.com This image cannot currently be displayed. The form of the equation is immaterial in a mathematical sense. But in CFD applications, success or failure often depends on in what form the equations are formulated. This is a result of the CFD techniques not having firm theoretical foundation regarding stability and convergence, von Neumann’s stability analysis notwithstanding. Recall that von Neumann stability analysis is applicable only for linear PDEs. The Navier-Stokes equations are non-linear. Docsity.com ne ——®———— Control surface $ eee ee —— — eee ee a — Finite control volume fixed in space with the fluid moving through it (a) -F pe ee ee “—-7| Volume d¥ ee a —_———>—_____ =e oat ad Infinitesimal fluid element fixed in space with the fluid moving through it (b) FIG, 2.2 Finite control volume moving with the fluid such that the same fluid particles are always in the same control volume _av a. a / os <<. Infinitesimal fluid element moving along a streamline with the velocity V equal to the local flow velocity at each point Models of a flow. (a) Finite control volume approach; (b) infinitesimal fluid element approach. Docsity.com  This image cannot currently be displayed. Consider a differential volume element dV in the flow field. dV is small enough to be considered infinitesimal but large enough to contain a large number of molecules for continuum approach to be valid. dV may be: • fixed in space with fluid flowing in and out of its surface or, • moving so as to contain the same fluid particles all the time. In this case the boundaries may distort and the volume may change. Docsity.com This image cannot currently be displayed. Substantial derivative (time rate of change following a moving fluid element) Insert Figure 2.3 Docsity.com This image cannot currently be displayed. The time derivative can be written as shown on the RHS in the following equation. This way of writing helps explain the meaning of total derivative. 2 1 2 1 2 1 2 1 1 1 12 1 2 1 2 1 2 11 ...........(2.1)x x y y z z t t x t t y t t z t t t ρ ρ ρ ρ ρ ρ − ∂ − ∂ − ∂ − ∂     = + + +      − ∂ − ∂ − ∂ − ∂       Docsity.com This image cannot currently be displayed. We can also write 2 1 2 1 2 1 lim t t D t t Dt ρ ρ ρ → − ≡ − 2 1 2 1 2 1 lim t t x x u t t→ − ≡ − 2 1 2 1 2 1 lim t t y y v t t→ − ≡ − 2 1 2 1 2 1 lim t t z z w t t→ − ≡ − Docsity.com This image cannot currently be displayed. where the operator can now be seen to be defined in the following manner. ∴ ...........(2.2)D u v w Dt x y z t ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ D Dt .........(2.3)D u v w Dt t x y z ∂ ∂ ∂ ∂ ≡ + + + ∂ ∂ ∂ ∂ Docsity.com This image cannot currently be displayed. A simpler way of writing the total derivative is as follows: ............(2.7)d dx dy dz dt x y z t ρ ρ ρ ρρ ∂ ∂ ∂ ∂= + + + ∂ ∂ ∂ ∂ ............(2.8)d dx dy dz dt t x dt y dt z dt ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ..........(2.9)d u v w dt t x y z ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ Docsity.com This image cannot currently be displayed. The above equation shows that and have the same meaning, and the latter form is used simply to emphasize the physical meaning that it consists of the local derivative and the convective derivatives. Divergence of Velocity (What does it mean?) ( Section 2.4) Consider a control volume moving with the fluid. Its mass is fixed with respect to time. Its volume and surface change with time as it moves from one location to another. d dt ρ D Dt ρ Docsity.com This image cannot currently be displayed. Insert Figure 2.4 Docsity.com This image cannot currently be displayed. If we now make the moving control volume shrink to an infinitesimal volume, δv, the above equation becomes When 0 the volume integral can be replaced by on the RHS to get the following. The divergence of is the rate of change of volume per unit volume. ( )( ) ...............(2.13) V D V V dV Dt = ∇ ⋅∫∫∫ V∆ → VVδ∇ ⋅ V 1 ( ) .........(2.14)D VV V Dtδ ∇ ⋅ = Docsity.com =a FIG. 2.5 Finite control volume fixed in space. _ Docsity.com  This image cannot currently be displayed. Continuity Equation (2.5) Consider the CV fixed in space. Unlike the earlier case the shape and size of the CV are the same at all times. The conservation of mass can be stated as: Net rate of outflow of mass from CV through surface S = time rate of decrease of mass inside the CV Net rate of Docsity.com