NUMERICAL APPROACH
3.1 PREVIOUS WORKS ON THE LIQUID BRIDGE PROBLEM
With the development of powerful and easily accessible computers a lot of new and interesting results have been obtained numerically in the field of hydrodynamical stability of liquid bridges in the last decade. The first works aiming at investigation by means of numerical simulations the flow in cylindrical floating-zone melt were done by Chang and Wilcox [14]. Their results were flow pattern and temperature field simulated for silicone. The summary of the most significant recent results is given in the book of Kuhlmann [55]. By using linear stability analysis Wanschura et al. [152] established the stability diagram for a wide range of Prandtl numbers (0≤P r≤5) for the bifurcation from a steady axisymmetric flow to a time dependent one. The stability diagram was obtained, showing the dependence of the critical Reynolds number upon the Prandtl number. It consists of two branches belonging to the region of small ( 0≤P r≤0.07), and large (0.8 ≤ P r ≤ 5) Prandtl numbers. The mechanism of instability has been identified for each of them. The branch for higher Prandtl numbers corresponds to oscillatory instability with an azimuthal wave numberm= 2 (see its definition bellow in section 4.3). The first instability for small Prandtl numbers is stationary with the same wave numberm= 2. Levenstam and Amberg [64] showed that for higher Reynolds numbers this flow undergoes the second bifurcation, from a stationary to an oscillatory flow (see Fig. 2.3). The time-dependent flow above the secondary instability was analyzed in detail by Leypoldt et al. [68] and the mechanism of instability was discussed. In the gap between two branches 0.1 ≤ P r ≤ 0.8 Chen et al. [18] have found oscillatory instability with a larger azimuthal wave number m = 3. A more detailed study of the flow instability in this gap has been recently proposed by Levenstam et al. [65]. For the intermediate range 0.07-0.84 of Prandtl numbers they have found four new branches, with azimuthal wave numbers 2, 3, and 4, which all oscillate.
The type of the supercritical hydrothermal waves appeared after the Hopf bifurcation is presently under discussion in literature. The detailed analysis of wave properties made by Leypoldt et al. [68] for medium Prandtl number,P r = 4, has shown that the standing wave appeared near the onset of instability decays into a traveling wave. For large Prandtl number, close to the onset of instability, Carotenuto et al. [12] (P r = 74) and Shevtsova et al. [125]
(P r = 105) experimentally observed that the oscillatory pattern is a standing wave. Standing and traveling waves have been numerically obtained by Savino & Monti [103] for large Prandtl numbers P r = 30 and 74. They observed that immediately after the onset of instability, the oscillatory flow can be described by a standing wave with a pulsating temperature distribution.
3.2 DISCRETIZATION OF PARTIAL DIFFERENTIAL EQUATIONS (PDE’s). GENERAL REMARKS.
It is important to realize that there are many different approaches which enable calculation of approximate solutions of PDE’s. The real, physical situation has to be modeled mathematically and that will introduce approximations resulting in governing differential equations. Next the resulting differential equations from the model need to be resolved. For this, they must be dis- cretized introducing another set of approximations. Discretization eliminates another property of real physical system – spatio-temporal continuity. Finally the discrete system needs to be solved and again this is often only done approximately, either because physical constants may be known only approximately or because the discrete system has only an approximate numerical solution.
In this section, we examine how simple differential systems can be discretized and resolved in a number of ways, some of which reflect the physical basis of the equations. We do this via a simple model problem and a series of observations which extend the model problem to cover many aspects of numerical solution of PDE’s.
In describing the variation of physical quantities we use functions: displacement as a function of time and so on. Thus it is important to think about how functions come about. If we consider a simple example,y= sin(x), then y(x) could equally well be defined by
(a) geometrically by using a unit circle, (b) algebraically by using a power series,
(c) as the solution of a differential equation with following boundary conditions:
∂2y
∂x2 +y= 0, y(0) = 0, ∂y
∂x(0) = 1. (3.1)
Most of the functions we use in physics and engineering are defined only by the last method:
they are defined by a differential equation which are just the mathematical expression of conser- vation laws, and it is very common to end up with two space derivatives. Most such equations may be solved only numerically and there are no methods of finding full analytical solutions.
3.3 MODEL PROBLEM
A simple model problem needs to have a second derivative and a source term, both common in conservation laws. Let us consider an equation:
−∂2y
∂x2 +k2y=f(x), x∈[0,1], y(0) =y0, y(1) =yN. (3.2) Although the differential equation eq.( 3.2) can be solved analytically it needs to be treated as if a computed solution has to be found. There are a number of ways of discretizing this model equation.
3.3.1 Finite Differences
The finite difference approach is the most popular discretization technique, owing to its sim- plicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series. Consider the following Taylor expansions
y(x+ ∆x) =y(x) + ∂y
∂x∆x+ ∂2y
∂x2
∆x2 2 +∂3y
∂x3
∆x3
6 +O(∆x4), (3.3)
y(x−∆x) =y(x)− ∂y
∂x∆x+ ∂2y
∂x2
∆x2 2 −∂3y
∂x3
∆x3
6 +O(∆x4). (3.4) Thus, from eq.(3.3) for the first order derivative we get:
forward difference
∂y
∂x = y(x+ ∆x)−y(x)
∆x +O(∆x), (3.5)
backward difference
∂y
∂x = y(x)−y(x−∆x)
∆x +O(∆x), (3.6)
centered difference
∂y
∂x = y(x+ ∆x)−y(x−∆x)
2∆x +O(∆x2). (3.7)
An approximation for the second order derivative is obtained by adding eq.(3.3) + eq.(3.4):
∂2y
∂x2 = y(x+ ∆x)−2y(x) +y(x−∆x)
∆x2 +O(∆x2). (3.8)
The terms O(∆x) and O(∆x2) are the remainders which are truncated (truncation error) to obtain the approximate derivatives. The centered difference approximation given by eq.(3.7) is more precise than the forward difference eq.(3.5) or the backward difference because the truncation error is of higher order, a consequence of cancelation of terms of the expansions when taking the difference between eq.(3.3) and eq.(3.4). Since the centered difference involves both neighboring points, there is more balanced information on the local behavior of the function.
Until now, a regular grid with uniform mesh intervals was assumed. In many cases it may be necessary to discretize the equations on an irregular grid. Traditionally non-uniform grids are widely used for improving the accuracy of the approximate solution. However, the disadvan- tageous property of approximating second-order elliptic operators on non-uniform grids is that the second order of local approximation is lost (see, e.g., book by Samarskii [100]).
Figure 3.1: Non-uniform 1D mesh.
Consider a non-uniform discretization, for example (Fig. 3.1). If finite difference approxima- tions are sought at the grid pointi, we can use the following Taylor expansions:
y(x+ ∆xi) =y(x) +∂y
∂x∆xi+∂2y
∂x2
∆x2i 2 +∂3y
∂x3
∆x3i
6 +O(∆x4i), (3.9) y(x−∆xi−1) =y(x)− ∂y
∂x∆xi−1+∂2y
∂x2
∆x2i−1 2 − ∂3y
∂x3
∆x3i−1
6 +O(∆x4i−1). (3.10)
For the centered difference we will get:
∂y
∂x = y(x+ ∆xi)−y(x−∆xi−1)
∆xi−1+ ∆xi +O
∆x2i−1+ ∆x2i
∆xi−1+ ∆xi
. (3.11)
Due to the non uniformity of the mesh intervals, the terms corresponding to the second order derivatives in the Taylor expansions do not cancel exactly, therefore we have a larger truncation error.
The second order derivative:
∂2y
∂x2 = 2
y(x+ ∆xi)−y(x)
∆xi −y(x)−y(x−∆xi−1)
∆xi−1
/(∆xi−1+ ∆xi) +O
∆x2i + ∆x2i−1
∆xi−1+ ∆xi
. (3.12)
For simplicity reason we consider a uniform mesh everywhere below if not mentioned other- wise. Now if yi =y(xi), fi =f(xi), i = 0, ..., N; ∆x = 1/(N −1) =h then eq.(3.2) will get the following discretize form:
−yi+1−2yi+yi−1
h2 +k2yi=fi. (3.13)
If we let yT = (y0, ..., yN) be a vector of values at the points (x0, ..., xN), fT = (y0/h2, f1, ..., fN−1, yN/h2) (notice that y0 and yN are known), define matrices K, I∗
K =
1 0 0 0 . 0 0 0
−1 2 −1 0 . 0 0 0
0 −1 2 −1 . 0 0 0
0 0 −1 2 . 0 0 0
. . . . . . . .
0 0 0 0 . −1 2 −1
0 0 0 0 . 0 0 1
, (3.14)
I∗ =
0 0 0 0 . 0 0 0 0 1 0 0 . 0 0 0 0 0 1 0 . 0 0 0 0 0 0 1 . 0 0 0 . . . . . . . . 0 0 0 0 . 0 1 0 0 0 0 0 . 0 0 0
, (3.15)
Then the finite difference scheme of eq.(3.13) is written in a discrete version as 1
h2K+k2I∗
y =f. (3.16)
The known valuesy0, yN are present among ”unknowns” since if one of the boundary condi- tions involved a derivative (e.g. fluxes through the boundaries are specified) then keeping these values as ”unknowns” is needed.
In general case when the boundary conditions are defined as ∂y
∂x+y
1/2 =a,
∂y
∂x+y
N−1/2 =b, (3.17)
where the sub indexes 1/2 and N−1/2 mean that the points between 0 and 1, andN −1 and N respectively are under consideration, vector fT = (a, f1, ..., fN−1, b) and the matricesK, I∗ will be written:
K =
−2h 2h 0 0 . 0 0 0
−1 2 −1 0 . 0 0 0
0 −1 2 −1 . 0 0 0
0 0 −1 2 . 0 0 0
. . . . . . . .
0 0 0 0 . −1 2 −1
0 0 0 0 . 0 −2h 2h
, (3.18)
I∗=
1/k2 1/k2 0 0 . 0 0 0
0 1 0 0 . 0 0 0
0 0 1 0 . 0 0 0
0 0 0 1 . 0 0 0
. . . . . . . .
0 0 0 0 . 0 1 0
0 0 0 0 . 0 1/k2 1/k2
, (3.19)
3.3.2 Methods of solving linear systems
How to solve the matrix equations obtained as a result of descretizing the PDE’s? Methods used for obtaining the numerical solution of a linear system
Au¯= b
¯ (3.20)
are numerous and among them only those are chosen for applications that are best suited to the capability of computers. Methods are compared by CPU time needed for the final solution of demanded accuracy to be found. There was also a criteria of minimal storage of the data requirements and operational memory needed but nowadays it is not actual any more since computers’ capacity and RAM became really huge.
All the methods of solving the matrix equations (3.20) lie in two classes: direct (or elimination methods), and iterative (indirect methods). Iterative methods are usually used in case if theA matrix is a large sparse matrix. Iterative methods are based upon generating a sequence of u
¯
k
approximate solutions that finally converges to exact solution u
¯ = A−1b
¯. Direct methods are applied for problems with dense coefficient matrix.
Direct methods are compared for efficiency of on the basis of the number of arithmetical operations required.One of the methods based on Cramer’s rule turns to be very inefficient. The Cramer’s rule states that:
Cramer’s rule. In the system (3.20) of order m, letdetA= 0, and letAi (m = 1,..,m) be the determinant of the matrix obtained from A by replaying the i−th column of A by the vector b
¯. Then the solution of (3.20) is
ui = detAi
detA, i= 1, ..., m. (3.21)
This method is very inefficient because of evaluation of (m+1) determinants. For instance, if the order m = 10 it takes 68 million multiplications. The number of operations involved in this direct method is of order (m!). A more efficient method of calculating the determinants may decrease the number of operations down to 3000 multiplications, but it stays very large in comparison with Gaussian elimination, which requires about 380 multiplications.
Gaussian elimination is based on transformations of the system (3.20) into the upper trian- gular system. The number of transformations is (m-1).
Gauss-Jordan schemeleads finally to a diagonal matrix rather than triangular. The solution u¯ is obtained by dividing the components of the right-hand side by the corresponding diagonal elements of the transformed matrix.
Triangular, or LU, decompositionis another method of which the idea is to find two triangular matrices L and U (L is lower triangular and U is upper triangular) that gives A = LU.The amount of work is the same as for the Gaussian elimination.
3.3.3 Finite volume method
It has already been observed that many differential equations which we would like to solve come from conservation laws which are integrals over volumes. This idea has been carried onto the discretization of such equations by instead of interpreting yi as an approximation to a point value, y(xi), rather
Yi = 1 V V
ydv, (3.22)
whereV is a control volume (an interval in 1D case and surface in 2D one).
In one sense we now reverse the process by which we arrived at the differential equation from a conservation law: take the differential equation and integrate. For the considered 1D model equation eq.(3.2)
(i+1/2)h (i−1/2)h
(−∂2y
∂x2 +k2y−f(x))dx= 0, (3.23) then definingFi = h1((ii−1+1//2)2)hhf(x)dx
∂y
∂x(i−1/2)h−∂y
∂x(i+1/2)h +k2hYi−hFi = 0, (3.24) To represent the fluxes across the faces of the volume at (i−1/2)h and (i+ 1/2)h we can use the usual approach of centered differences in terms of the integral quantities Yi:
∂y
∂x(i+1/2)h = Yi+1−Yi 2h .
So that we end up with a coefficient matrix which is identical to the finite difference one we derived above, but the right hand side is now a vector of integrals off:
1
h2K+k2I∗
Y =F. (3.25)
While this appears very similar to an ordinary finite difference method, if using unstructured meshes in two and three space dimensions (so that the volumes are either arbitrary triangles or quadrilaterals) the finite volume method is much easier to apply than conventional finite differences.
3.3.4 Unsteady PDE’s
In applications quite often we are interested in dynamics of the system and thus unsteady equations should be solved. Consider another model equation:
∂y
∂t =ξ∂2y
∂x2 −k2y+f(x), (3.26)
x∈[0,1], y(x= 0) =y0, y(x= 1) =yN, y(t= 0) =y0(x).
A simple approach is to discretize the time derivative with a forward difference as
∂y
∂t = y(t+ ∆t)−y(t)
∆t . (3.27)
Then introducing a set of time pointstj =j∆t, j = 0,1, ...and an abbreviationyni =y(xi, tn) we will get descretized form of eq.(3.26):
yin+1−yni
∆t =ξymi+1−2yim+yim−1
h2 −k2yim+fi. (3.28) The super index m can be equal either n or n−1 depending on a scheme used. The solution is known at time tn and a new solution must be found at time tn+1. Starting from the initial condition at t= 0, the time evolution is constructed after each time step either explicitly, by direct evaluation of an expression obtained from the discretized equation, or implicitly, when solution of a system of equations is necessary.
Three different schemes exist:
(a) explicit scheme, m=n;
(b) implicit scheme, m=n+ 1;
(c)θ–method, when we approximate the right hand side (RHS) as a weighted sum of values at m=nand n+ 1: θRHSn+1+ (1−θ)RHSn.
Explicit scheme.
An explicit approach is readily obtained by substituting the space derivative with the 3-point finite difference evaluated at the current time step. If we approximate the RHS of the differential equation at timetn and letµ=ξ∆t/h2 then we have an equation defining yn+1
yni+1= (I−k2∆tI∗+µK)yin+ ∆tfi. (3.29) The vectoryn+1is given explicitly in terms of known quantities so there is no matrix problem to solve.
Implicit scheme.
A fully implicit scheme is obtained by discretizing the space derivative with the finite dif- ference approximation attn+1. Approximation of the RHS of the differential equation at time tn+1 gives
(I+k2∆tI∗−µK)yin+1 =yin+ ∆tfi, (3.30) and a matrix problem has to be solved at each time step.
θ–method.
It is a fairly general implicit scheme, a combination of the explicit and implicit methods resulting in equation
(I+θk2∆tI∗−θµK)yni+1= (I−(1−θ)k2∆tI∗+ (1−θ)µK)yin+ ∆tfi, (3.31) and again a matrix problem has to be solved at each time step.
The parameter θ∈(0,1) may be different. When θ= 0 the explicit scheme is recovered, if θ= 1/2 the scheme is called Crank-Nicholson.
Other schemes originate from an evaluation of the time derivative by a centered difference approximation, in the attempt to reduce the truncation error
yni+1=yni−1+ 2(−k2∆tI∗+µK)yni + 2∆tfi. (3.32) This scheme involves three time levels at each step of the iteration, but is explicit and is potentially much more accurate than the schemes above because of the more precise evaluation of the time derivative. Unfortunately, better truncation error is not a guarantee for the overall stability of the scheme. Three-level schemes need the knowledge of the solution at two consec- utive times tn−1 and tn to obtain the solution attn+1. It means that more memory is required as well as a special starting procedure.
3.3.5 Stability of iterative methods
The theory of stability of numerical schemes is a very useful and important and powerful tool for theoretical investigations of specific schemes. Instead of analyzing each particular difference scheme, it suffices to reduce it into the canonical form and apply the known results. The most important results on this subject can be found in the book by Samarskii and A.V. Goolin [101].
The survey papers of V. Thom´ee [143],[144] should also be mentioned here. The stability theory is a basis for the analysis of the convergence of an approximative solution to the exact solution, provided that the mesh width tends to zero.
We have seen that discretization of PDE’s leads naturally to a matrix equation for a vector of unknowns which might represent nodal values of a function, local averages of a function or coefficients in an expansion in terms of a set of basis functions. In most real problems the dimension of the coefficient matrix is so large that Gauss elimination is not a feasible option, so that an interactive method must be used to approximate the solution vector. We take as our model problem eq.(3.26).
Consider a general form of the discretized eq.(3.26)
yn+1=Gyn+F, (3.33)
where matrixG is
G=I−k2∆tI∗+µK f or explicit scheme, G= (I+k2∆tI∗−µK)−1 f or implicit scheme, G= (I+θk2∆tI∗−θµK)−1·(I −(1−θ)k2∆tI∗+ (1−θ)µK) f or θ−method.
By definition, the true solution ˜y of eq.(3.33) satisfies
˜
y=G˜y+F. (3.34)
so defining an error en=yn−y˜we have
en+1 =Gen=Gn+1e0. (3.35)
Thus the iteration will converge providedGn→0, ifn→ ∞.
Theorem. If Ghas eigenvaluesλr, the iteration converges as n→ ∞ provided the spectral radius of the matrix ρ(G) = maxr(λr)<1.
It can be very difficult to calculate all of the eigenvalues of the iteration matrix. One approach which has been very successful in providing practical stability criteria is to extend the domain of the PDE to the whole real line (or the whole plane ...) and then to look at stability of Fourier modes. On the real line, ifynj is assumed to behave like
ynj = [λ(k)]neikjh. (3.36)
then we can examine maxk|λ(k)|to decide on stability of the numerical scheme. It is important to realize that although practical problems do not have infinite domains, they do not have con- stant coefficients; nevertheless criteria which have come from Fourier analysis of model problems are usually very good predictors for the behavior of a numerical scheme [100, 101].
As an example consider heat equation
∂y
∂t =ξ∂2y
∂x2, (3.37)
Discretizing explicitely will give
yin+1=yni +µ(yni+1−2yin+yin−1), (3.38) then substituting the Fourier mode gives
λ= 1−4µ
sin(kh 2 )
2
, (3.39)
So that λis real and |λ(k)| ≤ 1 provided 0≤µ≤0.5, that is equivalent to ∆t≤ h22ξ. So, if the mesh is refined then the time step has to be reduced by a factor equal to the square of the refinement.
Discretizing implicitly will give
yin+1=yin+µ(yni+1+1−2yin+1+yin−1+1), (3.40) and substituting the Fourier mode gives
λ= 1
1 + 4µ
sin(kh2 )
2. (3.41)
In this case if µ is positive,λ≤1 and the scheme is always stable.
Being applied to the θ-method, Fourier analysis of stability gives:
λ=
1−4µ(1−θ)
sin(kh2 ) 2
1 + 4µθ
sin(kh2 )
2 , (3.42)
and the scheme is stable providedθ≥ 12− 41µ.
We have assumed that the largest growth rate modulus must be bounded by 1 for stability;
it is possible to conclude that there will be stability in the sense that solutions remain bounded, provided the largest eigenvalue ofGsatisfies
|λ| ≤1 +C∆t, as∆t→0 (3.43)
for some constant C >0, since then
|λ|n≤(1 +C∆t)n≤eCn∆t, (3.44) thus
|λ|n≤eCt, as n∆t→t if n→ ∞. (3.45) This is called Lax-Richtmyer stability. However for practical purposes, if the modulus of λ exceeds 1 for finite ∆t then the calculations invariably become too large to handle. Hence it is safer to require practical or strict stability, |λ| ≤1.
Consider an explicit discretization of a convection-diffusion equation,
∂y
∂t +a∂y
∂x =ξ∂2y
∂x2, (3.46)
yjn+1 =yjn−1
2ν(ynj+1−yjn−1) +µ(ynj+1−2ynj +yjn−1), (3.47) whereν = a∆ht. Using a Fourier mode
λ= 1−iνsin(kh)−4µ
sin(kh 2 )
2
, (3.48)
so if
|λ(k)|2 =
1−4µ
sin(kh 2 )
22 +4ν2
sin(kh
2 ) 2
1−
sin(kh 2 )
2
, (3.49) Lax-Richtmyer stability analysis gives, if 0≤µ≤0.5,
|λ(k)|2 ≤ a2µ
ξ ∆t, (3.50)
so that the scheme will be stable for ∆t→0 ifµ= ξ∆th2 has a fixed value. In practice this is not sufficiently strong, for instance withµ= 0.25, ν = 0.75|λ(k)|2 = 1 +174
sin(kh2 ) 2
−134
sin(kh2 ) 4
,
so that growth factors of greater than 1 occur for a range of small values of
sin(kh2 ) 2
leading to breakdown of a calculation.
If we write
|λ(k)|2 = 1−4(2µ−ν2)
sin(kh 2 )
2
+4(4µ2−ν2)
sin(kh 2 )
4
, (3.51)
then |λ(0)|2 = 1 and |λ(1)|2 = 1−4µ ≤ 1 provided 0 ≤ µ ≤ 0.5. Since we are just dealing with a quadratic all we need to have is the slope (with respect to
sin(kh2 )
2
) to be negative at sin(kh2 ) = 0 in order for |λ| ≤ 1 over the range 0 ≤
sin(kh2 )
2
≤ 1. Hence we also need 2µ−ν2 ≥0. This can be rearranged as
a2∆t2
2h2 ≤ ξ∆t h2 ≤ 1
4. (3.52)
The three methods of discretizing the unsteady PDE’s mentioned above (explicit, implicit andθ-method) are not the only ones utilized in practice. Table 3.1 describes the known schemes applied for the heat equation eq.( 3.37).
Thus, as a conclusion one should be in favor of implicit scheme of solving partial differential equations to make calculations faster and then more efficient.
3.4 NUMERICAL TECHNIQUE OF SOLVING THE SYSTEM OF NAVIER-STOKES EQUATIONS IN CYLINDRICAL GEOMETRY
A liquid bridge is considered, consisting of a fluid volume held between two differentially heated horizontal flat concentric disks of radiusR, separated by a distanced. See geometry in Fig. 3.2.
The temperaturesThotandTcold(Thot> Tcold) are prescribed at the upper and lower solid-liquid interfaces respectively, yielding a temperature difference of ∆T =Thot−Tcold. The free surface is considered cylindrical and non-deformable. The surface tension and density are taken as linear functions of temperature
σ(T) =σ(T0)−σT(T −T0), σT =−∂σ
∂T =const.
ρ=ρ0−ρ0β(T−T0) where β =−ρ−10 ∂ρ
∂T.
All other material properties are regarded as constant and the small temperature dependence of the density is taken into account in the gravity term only (the Boussinesq approximation).
The governing Navier-Stokes, energy and continuity equations in non-dimensional primitive- variable formulation in a cylindrical co-ordinate system.
∂vr
∂t + Γ r
∂
∂r(rvr2) +Γ r
∂
∂ϕ(vrvϕ) + ∂
∂z(vrvz) =−Γ∂p
∂r + +
∆vr−Γ2vr r2 −2Γ2
r2
∂vϕ
∂ϕ
+ Γvϕ2
r , (3.53)
∂vϕ
∂t + Γ r
∂
∂r(rvrvϕ) +Γ r
∂
∂ϕ(v2ϕ) + ∂
∂z(vϕvz) =−Γ1 r
∂p
∂ϕ +
