Momentum Heat Mass Transfer. MHMT2. Balance equations. Mass and momentum balances.
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Balance equations. Mass and momentum balances.
Fundamental balance equations. General transport equation, material derivative. Equation of continuity. Momentum balance - Cauchy´s equation of dynamical equilibrium in continua. Euler equations and potential flows. Conformal mapping.
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
Mechanics and thermodynamics are based upon the
Description of kinematics and dynamics of discrete mass points is recasted to consistent tensor form of integral or partial differential equations for velocity, temperature, pressure and concentration fields.
Transfer phenomena looks for analogies between transport of mass, momentum and energy. Transported properties are scalars (density, energy) or vectors (momentum). Fluxes are amount of passing through a unit surface at unit time (fluxes are tensors of one order higher than the corresponding property , therefore vectors or tensors).
Convective fluxes ( transported by velocity of fluid)
Diffusive fluxes ( transported by molecular diffusion)
Driving forces = gradients of transported properties
This table presents nomenclature of transported properties for specific cases of mass, momentum, energy and component transport. Similarity of constitutive equations (Newton,Fourier,Fick) is basis for unified formulation of transport equations.
Mass conservation principle can be expressed by balancing of a control volume (rate of mass accumulation inside the control volume is the sum of convective fluxes through the control volume surface). Analysis is simplified by the fact that the molecular fluxes are zero when considering homogeneous fluid.
Control volumes can be fixed in space or moving.The simplest case, directly leading to the differential transport equations, is based upon identification of fluxes through sides of an infinitely small FLUID ELEMENTfixed in space.
Mass flowrate through sides W and E
Mass conservation(fixed fluid element)
Using the control volume in form of a brick is straightforward but clumsy. However, tensor calculus is not necessary.
Using index or symbolic notation makes equations more compact
Continuity equation written in the index notation (Einstein summation is used)
Continuity equation written in the symbolic form (Gibbs notation)
Example: Continuity equation for an incompressible liquid is very simple
Observer (an instrument measuring the property ) can be fixed in space and then the recorded rate od change is
fixed observer measuring velocity of wind
Rate of change of property (t,x,y,z) recorded by the observer moving at velocity
Time changes of recorded by observer moving at velocity
Material derivative is a special case of the total derivative, corresponding to the observer moving with the particle (with the same velocity as the fluid particle)
observer in a balloon
intensity of inner sources or diffusional fluxes across the fluid element boundary
[Accumulation in FE ] + [Outflow of from FE by convection] =
This follows from the mass balance
These terms are cancelled
Integral balance of (fixed CV)
Integral balance in a fixed control volume has the advantage that it is possible to exchange a time derivative and integration operator(V is independent of time)
apply Gauss theorem (conversion of surface to volume integral)
Integral balance should be satisfied for arbitrary volume V
Therefore integrand must be identically zero
Remark: special case is the mass conservation for =1 and zero source term
and using this the differential balance can be expressed in the alternative form
Momentum balance = balance of forces is nothing else than the Newton’s law m.du/dt=F applied to continuous distribution of matter, forces and momentum.
Newton’s law expressed in terms of differential equations is called
valid for fluids and solids (exactly the same Cauchy’s equations hold in solid and fluid mechanics).
Momentum integral balance
MOMENTUMintegral balances follow from the general integral balances
Differential equations of momentum conservation can be derived directly from the previous integral balance
which must be satisfied for any control volume V, therefore also for any infinitely small volume surrounding the point (x,y,z) and
This is the fundamental result, Cauchy’s equation (partial parabolic differential equations of the second order). You can skip the following shaded pages, showing that the same result can be obtained by the balance of forces.
Cauchy’s equation holds for solid and fluids (compressible and incompressible)
formulation with primitive variables,u,v,w,p. Suitable for numerical solution of incompressible flows (Ma<0.3)
Ma-Mach number (velocity related to speed of sound)
Making use the previously derived relationship the Cauchy’s equation can be expressed in form
conservative formulation using momentum as the unknown variable is suitable for compressible flows, shocks…. Passage through a shock wave is accompanied by jump of p,,u but (u) is continuous.
These formulations are quite equivalent (mathematically) but not from the point of view of numerical solution – CFD.
Inviscid flow theory of ideal fluids is very highly mathematically developed and predicts successfully flows around bodies, airfoils, wave motion, Karman vortex street, jets. It fails in the prediction of drag forces.
Eulers’s equations are special case of Cauchy’s equations for inviscible fluids (therefore for zero viscous stresses)
Vorticity vector describes rotation of velocity field and is defined as
for example the z-coordinate of vorticity is
Using vorticity the Euler equation can be written in the alternative form
this formulation shows, that for zero vorticity the Euler’s equation reduces to Bernoulli’s equation: acceleration+kinetic energy=pressure drop+external forces
Proof is based upon identity: see lecture 1.
Inviscid flows are frequently solved by assuming that velocity fields and volumetric forces f can be expressedas gradients of scalar functions (velocity potential)
Vorticity vector of any potential velocity field is zero (potential flow is curl-free) because
to understand why, remember that for the Levi Civita tensor holds imn= -inm
Velocities defined as gradients of potential automatically satisfy Kelvins theorem stating that if the fluid is irrotational at any instant, it remains irrotational thereafter (holds only for inviscible fluids!).
Because vorticity is zero the Euler equation is simplified
… integrating along a streamline gives Bernoulli’s equation
In 2D flows it is convenient to introduce another scalar function, stream function
Velocity derived from the scalar stream function automatically satisfies the continuity equation (divergence free or solenoidal flow) because
Curves =const are streamlines, trajectories of flowing particles. For example solid boundaries are also streamlines. Difference is the fluid flowrate between two streamlines.
Advantages of the stream function appear in the cases that the flow is rotational due to viscous effects (for example solid walls are generators of vorticity). In this case the dynamics of flow can be described by a pair of equations for vorticity and stream function
In this way the unknown pressure is eliminated and instead of 3 equations for 3 unknowns ux uy p it is sufficient to solve 2 equations for and .
Problem of inviscid incompressible flows can be reduced to the solution of two Laplace equations for stream and potential functions, satisfying boundary conditions of impermeable walls ( ) and zero vorticity at inlet/outlet ().
Euler’s Equations vorticity and stream function
Let us summarize:
For incompressible (divergence-free) flows the velocity potential distribution is described by the Laplace equation (ensures continuity equation)
For irrotational (curl-free) flow the stream function should also satisfy the Laplace equation
Euler’s Equations flow around sphere
Example: Velocity field of inviscid incompressible flow around a sphere of radius R is a good approximation of flows around gas bubbles, when velocity slips at the sphere surface. Velocity potential can be obtained as a solution of the Laplace equation written in the spherical coordinate system (r,,)
Velocity potencial satisfying boundary condition at r and zero radial velocity at surface is
The solution is found by factorisation to functions depending on r and on only
and velocities (gradient of )
Velocity profile at surface (r=R) determines pressure profile (Bernoulli’s equation)
Euler’s Equations flow around cylinder
Example: Potencial flow around cylinder can be solved by using velocity potencial function or by stream function. Both these functions have to satisfy Laplace equation written in the cylindrical coordinate system (the only difference is in boundary conditions).
Stream function satisfying boundary condition at r (uniform velocity U) and constant at surface is
see the result obtained by using complex functions
giving radial and tangential velocities
Compare with the previous result for sphere: the velocity decays with the second power of radius for cylinder, while with the third power at sphere (which could have been expected).
Euler’s Equations and complex functions
Many interesting solutions of Euler’s equations can be obtained from the fact that the real and imaginary parts of ANY analytical function
satisfy the Laplace equation (see next page).
z=x+iy is a complex variable (i-imaginary unit) and w(z)=(x,y)+i(x,y) is also a complex variable (complex function), for example
This is important statement: Quite arbitrary analytical function describes some flow-field. Real part of the complex variable w is velocity potential and the imaginary part Im(w) is stream function!
Simple analytical functions describe for example sinks, sources, dipoles. In this way it is possible to solve problems with more complicated geometries, for example free surface flows, flow around airfoils, see applications of conformal mapping.
=const Equipotential lines
Derivative dw/dz of a complex function w(z=x+iy)=+i with respect to z can be a complex analytical function as soon as both Re(w), Im(w) satisfy the Laplace equation
Result should be independent of the dx, dy selection, therefore
and this requirement is fulfilled only if both functions , satisfy Cauchy-Riemann conditions
The real and the imaginary part of derivative dw/dz determine components of velocity field
Euler’s Equations and complex functions
Example: Let’s consider the transformation w(z)=az2 in more details
The same graph can be obtained from inverse transformation z(w)
Example: Potential flow around cylinder with circulation can be assumed as superposition of linear parallel flow w1(z)=Uz, dipole w2(z)=UR2/z and potential swirl w3(z)=/(2i) ln z (see the previous table).
Substituting coordinates x,y by radius r and angle results into (x+iy=r ei)
Comparing real and imaginary part potential and stream functions are identified
velocity potential is the real part of the analytical function w(z)
stream function is the imaginary part of the analytical function w
without circulation, I have a problem in Matlab
Example: Potential flow around elliptic cylinder. Previous example solved the problem of potential flow around a cylinder with radius R, described by the conformal mapping
The analytical function transforming outside of an elliptical cylinder to the plane of complex potential w= +i can be obtained in two steps: First step is a conformal mapping (z) transforming ellipse with principal axis a,b to a cylinder with radius a+b. The second step is substitution of the mapping (z) to the velocity potential
There exist many techniques how to identify the conformal mapping (z) transforming a general closed region in the z=x+iy plane into a unit circle, for example numerically or in terms of Laurent series
…this is the way how to solve the problems of flow around profiles, for example airfoils. It is just only necessary to find out a conformal mapping transforming the profile to a circle.
For the conformal mapping of ellipse only three terms of Laurent’s series are sufficient
Inversion mapping (z) is the solution of quadratic equation
Complex potential (potential and stream function) is therefore
Generally speaking it does not matter if we select analytical function w(z) mapping the spatial region (z=x+iy) to complex potential region w=+i, or vice versa.
This is because inverse mapping is also conformal mapping.
Please notice the fact, that in this case the role of z and w is exchanged, complex variable w is spatial coordinates x,y, while z=+i is complex potential of velocity field.
Solution for h=1 by MATLAB
See M.Sulista: Analyza v komplexnim oboru, MVST, XXIII, 1985, pp.100-101.
Disadvantage of the approach using stream function, complex variables and conformal mapping is its limitation to 2D flows. While in the 3D flow the irrotational velocity field can be described by only one scalar function , description of 3D solenoidal field (satisfying continuity equation) by stream function is not so simple. It is necessary to use a generalized stream function vector and to decompose velocities into curl free and solenoidal components (dual potential approach)
Divergence free (solenoidal flow)
Curl free (potential flow)
Vorticity vector is expressed in terms of the stream function vector
The dual potential approach increases number of unknowns (3 stream functions and 2 vorticity transport equations are to be solved) and is not so frequently used.
You should know what is it material derivative
Balancing of fluid particle Balancing of fixed fluid element
Reynolds transport theorem
Continuity equation and Cauchy’s equations
What is it vorticity, stream function and velocity potential
Special case for 2D flows
Complex potential, analytical functions and conformal mapping