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CEE 262A H YDRODYNAMICS. Lecture 5 Conservation Laws Part I. Conservation laws.

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## CEE 262A H YDRODYNAMICS

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**CEE 262A HYDRODYNAMICS**Lecture 5 Conservation Laws Part I**Conservation laws**The ultimate goal is to use Newton's 2nd law (F=ma) to relateforces on fluid parcels to their acceleration. A natural way to do this is to compute the equations of motion following a volume of fluid; We first begin with the governing equations for a solid, non-deformable object with volumeV, surface area A, and density r: V A Mass of object: Momentum of object:**Newton's 2nd law for the object is given by:**Surface forcese.g. Friction Body forcese.g. Gravity Example: Block of mass m pushed with force Falong surface with friction coefficient b: u F g m N**velocity of boundary**The problem for fluids is that the volume is generally not fixed in time, and so mass and momentum may leave the volume unless a material volume is employed. The key to understanding what to do is Leibniz's Theorem: Consider a volume V(t) for which the bounding surface moves (butnot necessarily at the fluid velocity) Which in 3D becomes:**Fixed volume–VF : Flow of fluid through system boundary**(control surface) is non zero, but velocity of boundary is zero. For this case we getMaterial Volume–VM : Consists of same fluid particles and thus the bounding surface moves with the fluid velocity. Thus, the second term from the Leibnitz rule is now non-zero, soUsing Gauss' theorem: This is Reynolds transport theorem, where D/Dt is the same as d/dtbut implies a material volume.**Note that the Reynolds transport theorem is often written in**themore general form which does not assume that the control volumeis bounded by a material surface. Instead, the control volume isassumed to move at some velocity and that of the fluid is defined as relative to the control volume, such thatIn this case, V is not necessarily a material surface. If ur=0, thenub=u and we revert to the form on the previous page. ~ ~ ~**The general form of all conservation laws that we will use**is: 1 Note: momentum is a vector quantity**Conservation of Mass**For any arbitrary material volume Mass is conserved (non-relativistic fluid mechanics) Since integral is zero for any volume, the integrand must be zero Process: We have taken an integral conservation law and used it to produce a differential balance for mass at any point**However,**and Thus if the density of fluid particles changes, the velocity field must be divergent. Conversely, if fluid densities remain constant,**But**Any other fluid property (scalar, vector,.. also drop triple integral)**Net Applied Force = Mass Acceleration**Why is this important/useful? Rate of Change of Momentum = Net Applied Force Because Newton’s 2nd law: But from above: Independent of volume type!**Some Observations**1. Incompressible [ No volumetric dilatation, fluid particle density conserved] Differential form of “Continuity”**Allows us to treat fluid as if it were slightly**incompressible 2. Slightly Compressible • Typically found in stratified conditions where Perturbation density due to motion (typ. 0.1-10 kg/m3 for water) Reference density (1000 kg/m3 for water) Background variation (typ. 1-10 kg/m3 for water) • Boussinesq Approximation • Vertical scale of mean motion << scale height • or Note: Sound and shock waves are not included !**Informal “Proof”**If a fluid is slightly compressible then a small disturbance caused by a change in pressure, , will cause a change in density . This disturbance will propagate at celerity, c. • If pressure in fluid is “hydrostatic” Now and [ Streamline curvature small]**Conservation of Momentum – Navier-Stokes**We have: • Two kinds of forces: • Body forces • Surface forces • Two kinds of acceleration: • Unsteady • Advective (convective/nonlinear) • Two kinds of surface forces: • Those due to pressure • Those due to viscous stresses Divergence of Stress Tensor**Plan for derivation of the**Navier Stokes equation • Determine fluid accelerations from velocities etc. (done) • Decide on forces (done) • Determine how surface forces work : stress tensor • Split stress tensor into pressure part and viscous part • Convert surface forces to volume effect (Gauss' theorem) • Use integral theorem to get pointwise variable p.d.e. • Use constitutive relation to connect viscous stress tensor to strain rate tensor • Compute divergence of viscous stress tensor (incompressible fluid) • Result = Incompressible Navier Stokes equation**Forces acting on a fluid**a) Body Forces: - distributed throughout the mass of the fluid and are expressed either per unit mass or per unit volume - can be conservative & non-conservative Force potential • Examples: • force due to gravity (acts only in negative z direction) • force due to magnetic fields We only care about gravity**b) Surface forces: - are those that are exerted on an area**element by the surroundings through direct contact - expressed per unit of area - normal and tangential components**c) Interfacial forces:**• - act at fluid interfaces, esp. phase discontinuities (air/water) • do not appear directly in equations of motion (appear as boundary conditions only) • e.g. surface tension – surfactants important • very important for multiphase flows (bubbles, droplets,. free surfaces!)**Very important deviation from text!!!**CEE262a (and most others) Full stress Deviatoric (viscous) stress Kundu and Cohen Full stress Deviatoric (viscous) stress**Stress at a point**(From K&C – remember difference in nomenclature,i.e. tij ← sij)**What is the force vector I need to apply at a face defined**by theunit normal vector to equal that of the internal stresses? Consider a small (differential) 2-D element cut away**Defining the stress tensor to be**d force component in x1 direction [ has magnitude of 1] And in general**“ Surface force per unit area”**(note this is a 2D area) But [see Kundu p90] Total, or net, force due to surface stresses**Conservation of momentum**Dimensions:dx1. dx2. dx3**Defining i component of surface force per unit volume to be**Sum of surface forces in x1 direction:**In general :**Force = divergence of stress tensor “Cauchy’s equation of motion”**But**Important Note: This can also be derived from the Integral From of Newton’s 2nd Law for a Material Volume VM and**Constitutive relation for a Newtonian fluid**“Equation that linearly relates the stress to the rate of strain in a Newtonian Fluid Medium” • Static Fluid: - By definition cannot support a shear stress • - still feels thermodynamic pressure • (in compression) (ii) Moving Fluid: - develops additional components of stress (due to viscosity) Hypothesis Note difference from Kundu ! Deviatoric stress tensor [Viscous stress tensor]**Assume**4th order tensor (81 components!) that depend on thermodynamic state of medium If medium is isotropic and stress tensor is symmetric only 2 non-zero elements of which gives or See derivation of l in Kundu, p 100**(i) Incompressible**(ii) Static Special cases In summary Cauchy's equation Constitutive relation fora compressible, Newtonian fluid.**Navier-Stokes equation**The general form of the Navier-Stokes equation is given by substitutionof the constitutive equation for a Newtonian fluid into the Cauchy equation of motion: Incompressible form (ekk=0):**Assuming**where**If “Inviscid”**Euler Equation Or in vector notation Inertia Pressure gradient Gravity (buoyancy) Divergence of viscous stress (friction)

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