- Email: [email protected]
Dynamic response of liquid curtains to time-dependent pressure fluctuations
Dynamic response of liquid curtains to timedependent pressure fluctuations J. I. Ramos Department Pennsylvania,
of Mechanical USA
Engineering,
Carnegie-Mellon
University,
Pittsburgh,
A domain-adaptive finite difference method is used to study the dynamic response of incompressible, isothermal, inviscid liquid curtains that are subject to imposed pressure jluctuations. The domainadaptive method involves a mapping that transforms the time-dependent physical domuin into a fixed computational domain und requires that the unknown location of the downstream boundary of the physical domain be determined by means of an ordinury differential equation that ensures global mass conservation. An iterative block-implicit method has been used to solve the finite difjjerence equations, and calculations are presented to illustrate the effects of the Froude number, convergence parameter, nozzle exit angle, amplitude and frequency of the imposed pressure fluctuations, and nozzle exit thickness-to-radius ratio on the liquid curtain convergence length. It is shown that the amplitude of the convergence length oscillations decreases as the Froude number, convergence parameter, jirequency of the imposed pressure oscillations, and nozzle exit thickness-to-radius ratio are increased and increases as the nozzle exit ungle and amplitude of the imposedpressurefluctuations are increased. The frequency of the convergence length oscillations is only a function of the Froude number and frequency of the imposed pressure fluctuations. It is also shown that the criticul value of the pressure coefficient determinedfrom analytical solutions of the steady-state equutions can be exceeded without affecting the stability and convergence of the liquid curtain if the frequency of the imposed pressure oscillations is sufficiently high. Keywords:
liquid curtains,
adaptive
methods,
finite difference
method,
block-implicit
methods,
forced
dynamical systems Introduction Liquid curtains are annular jets that are subject to surface tension and pressure differences that can converge on their symmetry axis under certain conditions. When they do converge, liquid curtains form enclosed volumes that can be used as chemical reactors for the direct reduction of zirconium from zirconium tetrachloride and sodium, scrubbing of radioactive and nonradioactive particulates, stack emission scrubbing for pollution control, and the like.’ Annular liquid curtains can also be used as protection systems in inertial confinement laser fusion reactors and for the determination of the surface tension of liquids. Analytical and numerical solutions of the steadystate equations governing the fluid mechanics of liquid curtains were obtained by Ramos’ as a function of the Froude number, convergence parameter, nozzle exit angle, pressure difference across the liquid curtain, and nozzle exit thickness-to-radius ratio. The analytical Address reprint requests to Dr. Ramos at the Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA 15213, USA. Received
126
26 September
1989; accepted
Appl. Math. Modelling,
31 August
1991,
1990
Vol. 15, March
studies presented in Ref. 2 are based on the assumptions that the slope and curvature of the liquid are small; the curtain is incompressible, inviscid, isothermal, and torsionless; and the gases surrounding and enclosed by the curtain are incompressible, inviscid, and isothermal. These analytical studies are in very good agreement with the numerical and experimental results reported in Ref. 3, in which the domains of validity of the analytical solutions were determined as a function of the liquid curtain parameters and asymptotic solutions for the convergence time were also obtained. Ramos and Pitchumani4 performed an analysis of the boundary layer along the outer surface of the liquid curtain and showed that friction effects are negligible at high Reynolds numbers and for small values of the nozzle exit angle. The analytical and numerical studies reported in Refs. 2-4 are valid for steady-state, impermeable liquid curtains and neglect mass absorption. Ramos and Pitchuman? have shown that at high Peclet numbers the mass absorption by the liquid curtain can be neglected and that the assumption of a constant pressure in the volume enclosed by the liquid curtain is justifiable. Ramos6 has shown that mass absorption by liquid curtains in zero-gravity environments can also be
0 1991 Butterworth-Heinemann
Unsteady liquid curtains: J. I. Ramos neglected if the Peclet number is sufficiently high and obtained analytical solutions for the mass absorption rate as a function of the Weber number, pressure difference across the liquid curtain, nozzle exit angle, and curtain thickness-to-radius ratio at the nozzle exit. Experimental studies7 have shown that the liquid curtain may fluctuate as a function of time for a certain range of the curtain parameters. In particular, oscillations were observed at high Froude numbers. These experimental data were obtained at relatively low Peclet numbers, and mass absorption by the liquid curtain was important. Other possible explanations for the experimentally observed fluctuations are turbulence at the nozzle exit, fluctuations in the mass flow rate at the nozzle exit, and pressure fluctuations in the gases surrounded by the liquid curtain. It must be noted that mass absorption by the liquid curtain is not negligible at low Peclet numbers and that gases must be injected into the volume enclosed by the curtain to prevent its collapse. The gas injection rate must be identical to the gas absorption rate by the liquid curtain to achieve a steady-state operation. If there are fluctuations in the gas injection rate, the curtain will fluctuate. Motivated by the experimental observations of unsteady liquid curtains, Ram09 integrated the equations governing the fluid dynamics of incompressible, isothermal, torsionless liquid curtains; used kinematic and stress conditions at the liquid curtain interfaces; and obtained a set of asymptotic equations for the liquid curtain mean radius, thickness, and axial and radial momentum. These equations are exact for liquid membranes, that is, for liquid curtains of zero thickness. For steady-state inviscid liquids and for inviscid gases surrounding and enclosed by the curtain, the equations derived in Ref. 8 coincide with those reported in Refs. 2-6. Furthermore, the equations derived in Ref. 8 can also be used to determine the unsteady dynamics of liquid curtains and, in particular, the curtain response to imposed pressure fluctuations, oscillations in the mass flow rate at the nozzle exit, fluctuations in the mass flow rate of the gases injected into the volume enclosed by the liquid curtain, and the collapse of a liquid curtain due to gas absorption. As a first step in explaining the experimentally observed oscillations in liquid curtains, this paper analyzes the dynamic response of curtains subject to imposed pressure fluctuations. Further studies will consider the dynamic response of liquid curtains when the mass flow rate at the nozzle exit and the mass injection rate into the volume enclosed by the liquid curtain fluctuate as a function of time. The unsteady response of liquid curtains subject to imposed pressure fluctuations has been studied by Ramos and Pitchumani,Y who used a fixed grid in their computations. However, since the liquid curtain convergence length, that is, the axial location at which the liquid curtain converges on the symmetry axis, is a function of time and must be determined from the condition that the inner radius of the liquid curtain at the convergence point is zero, a fixed grid does not employ
all the grid points in the calculations and requires that an estimate of the largest convergence length for the largest pressure difference across the liquid curtain be known before the calculations are performed. Otherwise, the convergence length may exceed the length of the computational domain. To eliminate the disadvantages associated with the use of a fixed grid, the present paper uses a domainadaptive method that maps the time-dependent physical domain into a fixed computational domain, employs all the grid points, and ensures global mass conservation. The domain-adaptive method presented in this paper determines the time-dependent convergence length from the solution of an ordinary differential equation that is coupled in a nonlinear and integral manner to the mass and linear momentum of the liquid curtain. The dynamic response of liquid curtains that are subject to imposed pressure fluctuations presented in this paper involves a study of forced, distributed parameter systems and is therefore connected with distributed dynamical systems. The paper is arranged as follows. The second section describes the problem formulation and the nondimensionalization of the governing equations. The third section introduces a domain-adaptive finite difference method. In the fourth and fifth sections the dynamic response of a liquid curtain that is subject to pressure oscillations and the conclusions, respectively, are presented. Problem formulation Consider the liquid curtain schematically shown in Figure I and assume that the liquid is incompressible, isothermal, and inviscid. Assume also that the gases enclosed by and surrounding the liquid curtain are incompressible, inviscid, and isothermal. Under these assumptions, one can integrate the Euler equations across the liquid curtain from r = R; to r = R, and apply Leibnitz’s rule to obtain the following system of equations:
(1) dmu amuu at+-= dz
mg + g
- (pi - p,) (2)
amv at+-=-
amuv
a2
where t is time; z is the axial location measured from the nozzle exit; II and v are the axial and radial velocity components of the liquid curtain, respectively; g is the gravitational acceleration; m = p Rb is the mass per radian and per unit length of the liquid curtain; p is the liquid density; m is the surface tension; R is the mean radius of the liquid curtain [R = (Ri + R,)/2]; b is the thickness of the liquid curtain (b = R, - R;);
Appl. Math. Modelling,
1991, Vol. 15, March
127
Unsteady
liquid curtains: J. I. Ramos
ferentiation and
with respect to z, for example, R’ = dR/dz; R’
J’ = (1 +
RR’R”
R'2)"2
-
(4)
(1 + R’2)3’2
At the liquid curtain mean radius the following kinematic condition must be satisfied: dR
aR
u=-+uat
Figure 1.
Schematic
Equation (1) is exact, equations (2) and (3) are asymptotic to O(b), and equation (10) is asymptotic to O(P). These equations are exact for inviscid, isothermal, incompressible liquid membranes, that is, for R = Ri = R, and b = 0. Multiplication of equation (5) by m and subsequent rearrangement of terms using the continuity equation (equation (I)) allow us to express the kinematic relationship at the liquid curtain mean radius as
of a liquid curtain
R, is the outer radius of the curtain;
R; is the inner radius of the curtain; pi is the pressure of the gases enclosed by the curtain; ,D~is the pressure of the gases surrounding the liquid curtain; the primes denote dif-
R”
z* = -2 RO
= ;
C
=(d:-,.., Pn
0
We = PVZR,, 2a
N* = m*v*
(l)-(3)
u* = 0UV gR,,
+
dmR -+
at
amRu = mu
Introducing
“*
(6)
az
the following nondimensional
groups:
vv
0
=
s&
.r=- @ V0
2&R,
N=F
Equations
b*
(5)
a2
Fr = -!$
m*=mN
R*R* )RW
R*’
RT = m*R*
J*'
M* = m*u”
PR?
0
(7)
= (1
+
RW2)"2
-
(1
+
&%')'I2
and (6) can be written as
aF*(u*) a2*
=G(U*)
(8)
where U* = (m*, RT, M*, N*)T F(U*) = (M*, RTM*Im*, G(U*)
(9)
M*‘lm*,
N*M*Im*)T
= [O, N*, (m*Fr - C,,,R*R*’
(10)
+ J*‘), (C,,R*
The superscript T denotes transpose; the primes denote differentiation with respect to z*; We is the Weber number; N is the convergence parameter; C,, is the pressure coefficient; Fr is the Froude number; T is a nondimensional time; the starred variables are dimensionless; and R, and V, denote the nozzle mean radius and total velocity at the nozzle exit, respectively. Note that U, = v, cos 0,
v, = V, sin 0,
(12)
where U, and u, denote the axial and radial velocity
128
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Math.
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15, March
- J*‘IR*‘)IT
(11)
components at the nozzle exit, respectively, and 8, is the angle that the tangent to the liquid curtain mean radius forms with the z-axis at the nozzle exit, that is, at2
= 0.
The boundary
m, = ,&>bo
conditions UC>=
at the nozzle exit are
V, cos 8,
0,
=
V, sin
e,, (13)
and the nondimensional boundary nozzle exit can be written as
conditions
at the
Unsteady
liquid curtains:
J. I. Ramos
T
(14)
where h,lR, is the liquid curtain thickness-to-radius ratio at the nozzle exit. Equation (8) represents a system of partial differential equations. Note that the left-hand side of equation (8) is a convection operator that involves only firstorder derivatives with respect to r and z*, whereas the right-hand side of the equations for M* and N* involve R*“, that is, a2R*ldz*‘. The values of pi and ,D<,and initial conditions must be specified in order to solve equation (8). In this paper it is assumed that the gases surrounding the liquid curtain are dynamically passive so that pt, is constant, that is, the liquid curtain falls in a surrounding medium of infinite extent. The value of pi depends on the gas absorption by the liquid curtain, gas mass injection in the volume enclosed by the curtain, mass flow rate fluctuations at the nozzle exit, and so on. At high Peclet numbers the gas absorption by the liquid curtain is small, and in the absence of fluctuations at the nozzle exit and if the liquid curtain is impermeable, pi can be related to the mass of gases enclosed by the liquid curtain as follows. If the gases enclosed by the liquid curtain are ideal, pi Vi = miR, T;
(15)
where Vi, m;, and T, denote the volume, mass, and temperature of the gases enclosed by the liquid curtain, respectively, and R, is the gas constant. In the absence of both mass injection into and gas absorption by the inner surface, that is, r = R;, of the liquid curtain, mj is constant. Furthermore, if T, is also assumed to be constant, then equation (15) can be written as pi = miRn TiI
vRf dz
(16)
where L denotes the liquid curtain convergence length, that is, the axial distance measured from the nozzle exit at which RJL, t) = 0. This condition determines L = L(t), which is not known a priori. Note that Ri = R - b/2
R, = R f b/2
Equation (17) can be nondimensionalized R,+ = R* - b*/2
R,* = R* + b*/2
(17) as (18)
where RT = RiIR,
b* = blR,
R: = RJR,
(19)
Furthermore, according to the nondimensionalization of m = pRb introduced in equation (7), b* = m*I(R*N)
= m*21(RfN)
integrodifferential equations whose downstream boundary, that is, z, = L, is not known. In this paper we will assume that C,,,,, that is, (pi - p,) is a specified function of time, so that equation (8) represents a forced distributed parameter system; therefore equation (16) will not be used. The reasons for this seemingly academic study is to determine the dynamic response of liquid curtains to imposed pressure fluctuations. Future studies will consider equation (16), that is, cases in which there are velocity or mass flow rate fluctuations at the nozzle exit and unsteady mass injection into the volume enclosed by the liquid curtain. Furthermore, analytical studies for long liquid curtains, that is, L >> R,,, have shown that for 0,, = O”, C,,,, > 1, the liquid curtain does not converge, that is, R(z, I) # 0. These analytical studies have also shown that if 0,, = 0” and C,,,, = 1, then R(z, t) = R,,, that is, a cylindrical annular jet is obtained. In this paper we will also determine whether the critical value of C,,,, = 1 for steady-state, long liquid curtains can be exceeded while ensuring that the convergence length is finite, that is, L < a. Furthermore, if this condition is satisfied and the convergence length fluctuates in time, the mass absorption rate (which is ignored in this paper) may be enhanced. Numerical
method
As was discussed in the previous section, equations (l)-(3) and (5) are a boundary value problem whose downstream boundary, that is, L” = LIR,,, is an unknown function of time that must be determined so that R*(L*, 7) = 0. Since the location of the downstream boundary is an unknown function of time, it is convenient to introduce the mapping (7, z*) -
(5717)
(21)
77= z*lL*(r)
(22)
where t=r
so that 0 I n 2 1, that is, in (5, q)-coordinates the convergence length corresponds to the fixed boundary given by 77 = 1, and the nozzle exit corresponds to n = 0. Introducing equations (21) and (22) into equation (S), we have
!$ + W*$ = G*(I/*)
(23)
where
(24)
(20)
Equation (16) and the definition of C,, in equation (7) indicate that equation (8) represents a system of
c~F*Ic?U* is a 4 unit matrix.
x
4 Jacobian matrix, and I is a 4 x 4
Appl. Math. Modelling,
1991, Vol. 15, March
129
Unsteady
liquid
J. I. Ramos
curtains:
Equation (23) can be discretized by using backward differences in time and upwind differences in space as
= A
+ Vi””
(25)
where A[ is the time step; AT is the grid spacing; the superscript n refers to time, for example, 5” = nA& and the subscript i denotes the axial location of the ith grid node. Equation (25) employs upwind differences for the convection terms and is O(A& A+accurate. Note that the algebraic multiplicity of the eigenvalue
tiplicity, that is, the number of linearly independent eigenvectors, is three. Note also that 0 i 775 1 and u* > dL*ldT, so that h > 0. This is the reason why upwind differences were employed to discretize the convection terms in equation (23). The term G* was di;;;tized by means of second-order accurate forEqiation (25) is a block bidiagonal system; the block dimension is 4 x 4. This system can be solved by forward substitution once L* and dL*ldr are known. To calculate L*, we consider the continuity equation, that is, the first equation of equation (8), (27)
(26) of the matrix H* is four, whereas its geometric
mul-
Equation (27) can be integrated from z” = 0 to 2* = L*(T) to yield, after application of Leibnitz’s rule,
(28) because m*(O, T)u*(O, 7) = M*(O, T) = N$
0
Fr
(2%
where equation (14) has been used. Substitution of equations (21) and (22) into equation (28) yields dL* -=&[$(L*[m*dq) dt
which can be written in finite difference L*“+l
=
(30)
+M*(l,t)-N$Fr]
- im*‘fdq]
m*“+‘(l,[)
form as L”” + [M*“+‘(l,t)
- N$Fr]
A,$}/(m*““(l,f)
0
0
- irn*‘l+ldq) 0
where the time derivatives have been discretized by means of backward differences. (31) Equation (25) was solved by means of an iterative block-implicit method as follows. Steady-state calculations were first performed for fixed values of b,,lR,,, C,,,,, N, I%, and Fr to determine the corresponding steady state values of r/f and L*. These values were used as initial guesses for UT”+’ and L*” and employed to solve equation (25) with dL*ldT = 0. The resulting values from the solution of equation (25) were UTkf’ and were used to determine Lfh+ ’ from equation (31). If
5
[(ui*k+’
U32]1/2
-
5
10
-4
(32)
i= I
where k denotes the kth interation and NP is the number of grid points used in the calculations, convergence was achieved within the time step; otherwise, an iterative procedure was used to solve equations (25) and (31). When performing steady-state calculations, that is, C,,, # Cpn(~), the following criterion was used to determine the steady state: jyi*n+’
130
AE>I -
UT”
2
“Z<
Appl. Math. Modelling,
,o_4
(33)
1991, Vol. 15, March
Note that ((ji*ll+’ - Ui*“)2 = (I/i*n+l - ui*“)(ui*n+’ - UT”)T
Presentation
(34)
of results
The transient calculations presented in this paper were performed with 401 grid points, that is, AT = 2.5 x 10e3, and At = 10P2. Calculations were also performed with smaller values of AT and At to determine that the results are independent of the time step and grid spacing. Steady-state calculations were also performed for C,, = 0 with the same values of AT and
Unsteady liquid curtains: J. I. Ramos
At to determine
the steady-state
convergence
where C,,, and w are the amplitude and angular frequency of the imposed pressure fluctuations, respectively. The dynamic response of liquid curtains to step and ramp pressure coefficients has also been studied by Ramos and Pitchumani’by means of nonadaptive finite difference methods. By introducing the nondimensional variables of equation (7), equation (35) can be written as
length
L&.
Calculations were performed to determine the dynamic response of liquid curtains subject to step, ramp, and sinusoidal changes in the pressure coefficient C,,, using the domain-adaptive technique presented in this paper as well as a fixed finite difference method. However, only the response of the liquid curtain to sinusoidal variations in the pressure coefficient will be considered in this paper, that is, C,, = C,,, sin (wf) Table 1.
C,,, = C,,,, sin (2~ St Fr 7) where St = wRJ(27rVJ
(35) Values of the parameters
Figure
Fr
N
2 3 4 5 6 7 8 9 10
3 3 3 3 3 Variable 5 3 5
625 625 625 625 625 625 Variable 625 625
(36)
is the Strouhal number.
used in the calculations &JR,
00 0 0 0 0 0 0 0 Variable 0
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Variable
c JJna 1 1 1 1 Variable 0.5 0.5 0.1 0.5
St 11120 1160 1I30 1115 1115 1130 1I30 1130 1I30
CPn
1x
--L ‘L,,
IO
I,
20
Figure 2.
Convergence
length as a function of time
Figure 4.
Convergence
length as a function of time
Figure 3.
Convergence
length as a function of time
Figure 5.
Convergence
length as a function of time
Appl.
Math. Modelling,
1991, Vol. 15, March
131
Unsteady liquid curtains: J. I. Ramos
Equations (8)-(1 I), (14), and (36) indicate that the dynamic response of liquid curtains subject to sinusoidal pressure fluctuations depends on Fr, N, $,, b,lR,, c pna, and St. The values of the parameters used in the calculations presented in this paper are shown in Table 1. Figures 2-5 show the convergence length normalized by the steady-state convergence length L.&, that is, the convergence length corresponding to St = 0, as a function of time and of the Strouhal number for a fixed value of C,,,. Figures 2-5 show that the liquid curtain does not respond instantaneously to the imposed pressure fluctuation and that there is a lag between L*(T) and C,,,(T). This lag is due to the inertia of the liquid curtain and depends on the Strouhal number. The larger the Strouha1 number, that is, the larger the frequency of the imposed pressure fluctuations, the smaller the lag time. For times larger than the lag time the convergence length responds rapidly to the imposed pressure fluctuations and exhibits a periodic behavior whose amplitude and frequency decrease and increase, respectively, as the Strouhal number is increased. The periodic character of the convergence length is to be expected, since only inviscid liquids are considered in this paper and there is no damping. Figures 2-5 also indicate that after the initial lag time, L* decreases as C,, increases, and L* increases as C,,, decreases. This is again due to the inertia of the liquid curtain. Furthermore, the amplitude of the oscillations in the convergence length increases as the Strouhal number is decreased; for St = 11120, L&,, is about 4 L& (Figure 2), whereas for St = l/15, L&, is about 1.5 L,*,. Analytical solutions of the steady-state equations governing the fluid dynamics of liquid curtains have shown that for 0, = 0” the liquid curtain converges for C,,, < 1, that is, L * is finite. However, L* is infinite for 8, = 0” and C,, 2 1. These analytical studies are based on a small curvature-small slope approximation and indicate that R* = 1, that is, a cylindrical annular jet is obtained, for 19,,= 0” and C,,, = 1. The results presented in Figures 2 and 5 correspond to Cpnr, = 1
4,
(
l.O-
z.o----
cpna’
3
3.0
-___-
3.5__-__
I
and indicate that for C,,, = 1 the liquid curtain converges even though a steady-state liquid curtain would not converge for C,,, = C,,,, = 1. The finite convergence length observed in Figures 2-5 indicates that the liquid curtain feels a pressure coefficient lower than C,,, due to the liquid inertia and lag time. Figures 2-5 also indicate that the higher the frequency of the imposed pressure fluctuations, the smaller the amplitude of the convergence length oscillations. In the limit St + 00 the amplitude of the convergence length oscillations tends to zero. Figure 6 shows the convergence length as a function of the amplitude of the imposed pressure fluctuations for a fixed value of the Strouhal number. This figure clearly illustrates that the frequency of the convergence length oscillations is independent of the Strouhal number, whereas the amplitude of the oscillations increases as CD,, is increased. This behavior is expected because of the fact that there is no damping. Figure 6 also shows that the amplitude of the imposed pressure fluctuations could be as high as the critical value of unity corresponding to steady-state liquid curtains. Furthermore, Figures 2-6 indicate that the value of C pnu could be further increased while still obtaining finite convergence lengths if the Strouhal number were increased. The results shown in Figures 2-6 correspond to fixed values of Fr, N, 8,,, and b,lR,. In Figures 7-10 the effects of these parameters on the liquid curtain response are examined for fixed values of C,,,,,,and St. Figure 7 illustrates the effects of the Froude number on the convergence length of the liquid curtain. As is indicated in equation (36), an increase in the Froude number implies an increase in the nondimensional frequency of the imposed pressure fluctuations. As a consequence, the frequency and amplitude of the convergence length oscillations increase and decrease, respectively, as the Froude number is increased. Note that the steady-state convergence length, that is, the convergence length corresponding to C,, = 0, increases as the Froude number is increased. Furthermore, the results shown in Figure 7 indicate that liquid
-’
9 *3
2
I
0 0 Figure 6.
132
IO Convergence
i
length as a function of time
Appl. Math. Modelling,
1991, Vol. 15, March
Figure 7. Convergence Froude numbers
length as a function of time for different
Unsteady liquid curtains: J. I. Ramos I.5
I
.D i lA 0.5
Figure 8. Convergence length as a function of time for different values of the convergence parameter
Figure 9. Convergence length as a function of time for different values of the nozzle exit angle
curtains with low Froude numbers are very sensitive to the imposed pressure oscillations, whereas curtains with high Froude numbers can withstand higher amplitudes of the imposed pressure oscillations. Figure 8 shows the convergence length as a function of the convergence parameter and clearly illustrates the effects of the liquid curtain inertia on the liquid curtain response. As is indicated in equation (7), the convergence number is the ratio of the inertial forces to surface tension for a fixed value of the Froude number, that is, the liquid curtain inertia increases in relation to the surface tension as the convergence parameter is increased. Therefore as the convergence parameter is increased, the amplitude of the convergence length oscillations decreases, and in the limit N+ m the amplitude of these oscillations tends to zero. Note also that the lag time between the imposed pressure oscillations and the oscillations of the convergence length increases as N is increased, but the frequency of the convergence length oscillations is independent of the convergence parameter because both St and Fr were fixed and there is no damping.
Figure 10. Convergence length as a function of time for several thickness-to-radius ratios
Figure 9 shows the convergence length as a function of the nozzle exit angle. Negative values of 0, correspond to flows directed towards the symmetry axis. Figure 9 indicates that the amplitude of the convergence length oscillations increases as 0, is increased. This behavior is to be expected, since positive values of 0, result in longer liquid curtains that are subjected to the imposed pressure oscillations over a larger surface area. Furthermore, the radius of the liquid curtain increases, whereas its thickness decreases, as 0, is increased. Note that the results shown in Figure 9 indicate that the frequency of the convergence length oscillations is independent of the nozzle exit angle. Figure IOpresents the convergence length as a function of nozzle gap width and indicates that the lag time increases as b,lR,, is increased, that is, as the liquid curtain inertia is increased. The amplitude of the convergence length oscillations increases as b,,lR,, is decreased, whereas their frequency is independent of the liquid curtain thickness-to-radius ratio at the nozzle exit. Conclusions The asymptotic equations governing the dynamics of liquid curtains subject to imposed pressure fluctuations have been solved numerically by means of a domainadaptive, iterative block-implicit method. The numerical method uses backward differences in time and upwind differences for the convection terms and results in a block bidiagonal system that can be solved by forward substitution. The main advantage of the domain-adaptive method presented in this paper is that the calculations are performed in a computational domain with fixed boundaries. However, an ordinary differential equation must be derived to determine the liquid curtain convergence length as a function of time. In the calculations presented in this paper that ordinary differential equation was derived from the continuity equation and ensures global mass conservation.
Appl.
Math. Modelling,
1991, Vol. 15, March
133
Unsteady
liquid curtains:
J. I. Ramos
Calculations were performed to determine the effects of the amplitude and frequency of the imposed pressure fluctuations, Froude number, nozzle angle, convergence parameter, and nozzle exit thickness-toradius ratio on the convergence length of the liquid curtain. These calculations show that the amplitude of the convergence length decreases as the frequency of the imposed pressure fluctuations, Froude number, convergence parameter, and nozzle exit thickness-toradius ratio are increased and increase as the nozzle exit angle and amplitude of the imposed pressure fluctuations are increased. The frequency of the convergence length oscillations increases as the Froude number and the frequency of the imposed pressure fluctuations are increased, but they are independent of the nozzle exit angle, convergence parameter, amplitude of the imposed pressure fluctuations, and nozzle exit thickness-to-radius ratio. It has also been shown that the critical value of the pressure coefficient determined from analytical solutions of the steady-state equations governing the fluid dynamics of liquid curtains can be exceeded if the curtain is subjected to pressure oscillations of sufficiently high frequency. This implies that stable liquid curtains can be obtained if the imposed pressure oscillations have high frequency. The calculations also show that the initial response of the liquid curtain is slow for high values of the nozzle exit thickness-to-radius ratio and convergence parameter and for small values of the Froude number and frequency of the imposed pressure oscillations.
m
mass per radian and per unit length of the curtain m = pRb dimensionless axial momentum per unit length and per radian convergence parameter, N = We/F? nondimensional radial momentum per unit length and per radian number of grid points hydrostatic pressure radial coordinate mean radius of curtain
m” N N* NP P k RT St
RT = m*R*
Strouhal number, St = wR,l(2rrV,) time axial component of the velocity of the liquid curtain four-dimensional vector defined in equation (9) radial component of the velocity of the liquid curtain velocity of liquid curtain, V = (u* + v*)“* Weber number, We = pV;R,,I2u axial coordinate
t U
u*
V
V
We 2
Greek symbols step size nondimensional axial coordinate defined in equation (22) eigenvalue angle between the tangent to the liquid curtain mean radius and the z-axis nondimensional time defined in equation (22) density of the liquid curtain surface tension of the liquid dimensionless time angular frequency of the imposed pressure fluctuations
Acknowledgments This work was supported by the Oftice of Basic Energy Sciences, U.S. Department of Energy, under Grant No. DE-FG02-86ER13.597 with Dr. Oscar P. Manley as technical monitor. This support is deeply appreciated. The author also appreciates the support provided by CRAY Research, Inc., through grants from the 1988 and 1989 CRAY Research and Development Grant Program, and by the Pittsburgh Supercomputing Center.
Nomenclature b
C Pn c PflU Fr F*( U*)
g G( V*) H*
Z J’ L
134
max 0 ss I
liquid curtain thickness pressure coefficient defined in equation (7) amplitude of oscillation of C,,, Froude number, Fr = V2/gRo four-dimensional vector defined by equation (IO) gravitational acceleration four-dimensional vector defined by equation (11) 4 x 4 matrix defined in equation (24) unit matrix term defined in equation (4) convergence length of the liquid curtain
Appl. Math. Modelling,
Subscripts e i
1991, Vol. 15, March
* T
outer surface of the liquid curtain inner surface of the liquid curtain maximum nozzle exit steady state corresponding to C,, = 0 Superscripts
differentiation coordinate dimensionless transpose
with respect to the axial quantity
References 1 2 3
Roidt, R. M. and Shapiro, Z. M. Liquid curtain reactor. Rept. No. 85M981, Westinghouse R&D Center, Pittsburgh, Pa., 1985 Ramos, J. I. Liquid curtains. I: Fluid mechanics. Chrm. Engrg. Sci. 1988, 43, 3171-3184 Ramos, J. I. Analytical, asymptotic and numerical studies of liquid curtains, and comparisons with experimental data. Appl. Math. Modding 1989, 14, 170-183
Unsteady liquid curtains: J. I. Ramos the Sixth Symposium on Energy Engineering Sciences: Flow und Transport in Continuu, U.S. Dept. of Energy, Argonne National Laboratory, Argonne, Illinois, May 1988, pp. 18-25
Ramos, J. 1. and Pitchumani, R. An analysis of laminar boundary layers on liquid curtains. J. Appl. Math. Phys. (ZAMP) 1989, 40, 72 l-739
Ramos, J. I. and Pitchumani, R. Gas absorption by liquid curtains. Rept. CO/88/3, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh, Pa., July 1988 Ramos, J. I. Annular liquid jets in zero-gravity. Appl. Muth. ModeUing 1990, 14, 630-640 Chigier, N., Ramos, J. I. and Kihm, K. Experimental and theoretical studies of vertical annular liquid jets. Proceedings of
8
9
Ramos, J. 1. Liquid membranes: Formulation and steady state analysis. Rept. CO/89/4, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh, Pa., February 1989 Ramos, J. I. and Pitchumani, R. Unsteady response of liquid curtains to time-dependent pressure oscillations. Rept. CO18814, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh. Pa., September 1988
Appl.
Math.
Modelling,
1991,
Vol. 15, March
135
of Mechanical USA
Engineering,
Carnegie-Mellon
University,
Pittsburgh,
A domain-adaptive finite difference method is used to study the dynamic response of incompressible, isothermal, inviscid liquid curtains that are subject to imposed pressure jluctuations. The domainadaptive method involves a mapping that transforms the time-dependent physical domuin into a fixed computational domain und requires that the unknown location of the downstream boundary of the physical domain be determined by means of an ordinury differential equation that ensures global mass conservation. An iterative block-implicit method has been used to solve the finite difjjerence equations, and calculations are presented to illustrate the effects of the Froude number, convergence parameter, nozzle exit angle, amplitude and frequency of the imposed pressure fluctuations, and nozzle exit thickness-to-radius ratio on the liquid curtain convergence length. It is shown that the amplitude of the convergence length oscillations decreases as the Froude number, convergence parameter, jirequency of the imposed pressure oscillations, and nozzle exit thickness-to-radius ratio are increased and increases as the nozzle exit ungle and amplitude of the imposedpressurefluctuations are increased. The frequency of the convergence length oscillations is only a function of the Froude number and frequency of the imposed pressure fluctuations. It is also shown that the criticul value of the pressure coefficient determinedfrom analytical solutions of the steady-state equutions can be exceeded without affecting the stability and convergence of the liquid curtain if the frequency of the imposed pressure oscillations is sufficiently high. Keywords:
liquid curtains,
adaptive
methods,
finite difference
method,
block-implicit
methods,
forced
dynamical systems Introduction Liquid curtains are annular jets that are subject to surface tension and pressure differences that can converge on their symmetry axis under certain conditions. When they do converge, liquid curtains form enclosed volumes that can be used as chemical reactors for the direct reduction of zirconium from zirconium tetrachloride and sodium, scrubbing of radioactive and nonradioactive particulates, stack emission scrubbing for pollution control, and the like.’ Annular liquid curtains can also be used as protection systems in inertial confinement laser fusion reactors and for the determination of the surface tension of liquids. Analytical and numerical solutions of the steadystate equations governing the fluid mechanics of liquid curtains were obtained by Ramos’ as a function of the Froude number, convergence parameter, nozzle exit angle, pressure difference across the liquid curtain, and nozzle exit thickness-to-radius ratio. The analytical Address reprint requests to Dr. Ramos at the Department of Mechanical Engineering, Carnegie-Mellon University, Pittsburgh, PA 15213, USA. Received
126
26 September
1989; accepted
Appl. Math. Modelling,
31 August
1991,
1990
Vol. 15, March
studies presented in Ref. 2 are based on the assumptions that the slope and curvature of the liquid are small; the curtain is incompressible, inviscid, isothermal, and torsionless; and the gases surrounding and enclosed by the curtain are incompressible, inviscid, and isothermal. These analytical studies are in very good agreement with the numerical and experimental results reported in Ref. 3, in which the domains of validity of the analytical solutions were determined as a function of the liquid curtain parameters and asymptotic solutions for the convergence time were also obtained. Ramos and Pitchumani4 performed an analysis of the boundary layer along the outer surface of the liquid curtain and showed that friction effects are negligible at high Reynolds numbers and for small values of the nozzle exit angle. The analytical and numerical studies reported in Refs. 2-4 are valid for steady-state, impermeable liquid curtains and neglect mass absorption. Ramos and Pitchuman? have shown that at high Peclet numbers the mass absorption by the liquid curtain can be neglected and that the assumption of a constant pressure in the volume enclosed by the liquid curtain is justifiable. Ramos6 has shown that mass absorption by liquid curtains in zero-gravity environments can also be
0 1991 Butterworth-Heinemann
Unsteady liquid curtains: J. I. Ramos neglected if the Peclet number is sufficiently high and obtained analytical solutions for the mass absorption rate as a function of the Weber number, pressure difference across the liquid curtain, nozzle exit angle, and curtain thickness-to-radius ratio at the nozzle exit. Experimental studies7 have shown that the liquid curtain may fluctuate as a function of time for a certain range of the curtain parameters. In particular, oscillations were observed at high Froude numbers. These experimental data were obtained at relatively low Peclet numbers, and mass absorption by the liquid curtain was important. Other possible explanations for the experimentally observed fluctuations are turbulence at the nozzle exit, fluctuations in the mass flow rate at the nozzle exit, and pressure fluctuations in the gases surrounded by the liquid curtain. It must be noted that mass absorption by the liquid curtain is not negligible at low Peclet numbers and that gases must be injected into the volume enclosed by the curtain to prevent its collapse. The gas injection rate must be identical to the gas absorption rate by the liquid curtain to achieve a steady-state operation. If there are fluctuations in the gas injection rate, the curtain will fluctuate. Motivated by the experimental observations of unsteady liquid curtains, Ram09 integrated the equations governing the fluid dynamics of incompressible, isothermal, torsionless liquid curtains; used kinematic and stress conditions at the liquid curtain interfaces; and obtained a set of asymptotic equations for the liquid curtain mean radius, thickness, and axial and radial momentum. These equations are exact for liquid membranes, that is, for liquid curtains of zero thickness. For steady-state inviscid liquids and for inviscid gases surrounding and enclosed by the curtain, the equations derived in Ref. 8 coincide with those reported in Refs. 2-6. Furthermore, the equations derived in Ref. 8 can also be used to determine the unsteady dynamics of liquid curtains and, in particular, the curtain response to imposed pressure fluctuations, oscillations in the mass flow rate at the nozzle exit, fluctuations in the mass flow rate of the gases injected into the volume enclosed by the liquid curtain, and the collapse of a liquid curtain due to gas absorption. As a first step in explaining the experimentally observed oscillations in liquid curtains, this paper analyzes the dynamic response of curtains subject to imposed pressure fluctuations. Further studies will consider the dynamic response of liquid curtains when the mass flow rate at the nozzle exit and the mass injection rate into the volume enclosed by the liquid curtain fluctuate as a function of time. The unsteady response of liquid curtains subject to imposed pressure fluctuations has been studied by Ramos and Pitchumani,Y who used a fixed grid in their computations. However, since the liquid curtain convergence length, that is, the axial location at which the liquid curtain converges on the symmetry axis, is a function of time and must be determined from the condition that the inner radius of the liquid curtain at the convergence point is zero, a fixed grid does not employ
all the grid points in the calculations and requires that an estimate of the largest convergence length for the largest pressure difference across the liquid curtain be known before the calculations are performed. Otherwise, the convergence length may exceed the length of the computational domain. To eliminate the disadvantages associated with the use of a fixed grid, the present paper uses a domainadaptive method that maps the time-dependent physical domain into a fixed computational domain, employs all the grid points, and ensures global mass conservation. The domain-adaptive method presented in this paper determines the time-dependent convergence length from the solution of an ordinary differential equation that is coupled in a nonlinear and integral manner to the mass and linear momentum of the liquid curtain. The dynamic response of liquid curtains that are subject to imposed pressure fluctuations presented in this paper involves a study of forced, distributed parameter systems and is therefore connected with distributed dynamical systems. The paper is arranged as follows. The second section describes the problem formulation and the nondimensionalization of the governing equations. The third section introduces a domain-adaptive finite difference method. In the fourth and fifth sections the dynamic response of a liquid curtain that is subject to pressure oscillations and the conclusions, respectively, are presented. Problem formulation Consider the liquid curtain schematically shown in Figure I and assume that the liquid is incompressible, isothermal, and inviscid. Assume also that the gases enclosed by and surrounding the liquid curtain are incompressible, inviscid, and isothermal. Under these assumptions, one can integrate the Euler equations across the liquid curtain from r = R; to r = R, and apply Leibnitz’s rule to obtain the following system of equations:
(1) dmu amuu at+-= dz
mg + g
- (pi - p,) (2)
amv at+-=-
amuv
a2
where t is time; z is the axial location measured from the nozzle exit; II and v are the axial and radial velocity components of the liquid curtain, respectively; g is the gravitational acceleration; m = p Rb is the mass per radian and per unit length of the liquid curtain; p is the liquid density; m is the surface tension; R is the mean radius of the liquid curtain [R = (Ri + R,)/2]; b is the thickness of the liquid curtain (b = R, - R;);
Appl. Math. Modelling,
1991, Vol. 15, March
127
Unsteady
liquid curtains: J. I. Ramos
ferentiation and
with respect to z, for example, R’ = dR/dz; R’
J’ = (1 +
RR’R”
R'2)"2
-
(4)
(1 + R’2)3’2
At the liquid curtain mean radius the following kinematic condition must be satisfied: dR
aR
u=-+uat
Figure 1.
Schematic
Equation (1) is exact, equations (2) and (3) are asymptotic to O(b), and equation (10) is asymptotic to O(P). These equations are exact for inviscid, isothermal, incompressible liquid membranes, that is, for R = Ri = R, and b = 0. Multiplication of equation (5) by m and subsequent rearrangement of terms using the continuity equation (equation (I)) allow us to express the kinematic relationship at the liquid curtain mean radius as
of a liquid curtain
R, is the outer radius of the curtain;
R; is the inner radius of the curtain; pi is the pressure of the gases enclosed by the curtain; ,D~is the pressure of the gases surrounding the liquid curtain; the primes denote dif-
R”
z* = -2 RO
= ;
C
=(d:-,.., Pn
0
We = PVZR,, 2a
N* = m*v*
(l)-(3)
u* = 0UV gR,,
+
dmR -+
at
amRu = mu
Introducing
“*
(6)
az
the following nondimensional
groups:
vv
0
=
s&
.r=- @ V0
2&R,
N=F
Equations
b*
(5)
a2
Fr = -!$
m*=mN
R*R* )RW
R*’
RT = m*R*
J*'
M* = m*u”
PR?
0
(7)
= (1
+
RW2)"2
-
(1
+
&%')'I2
and (6) can be written as
aF*(u*) a2*
=G(U*)
(8)
where U* = (m*, RT, M*, N*)T F(U*) = (M*, RTM*Im*, G(U*)
(9)
M*‘lm*,
N*M*Im*)T
= [O, N*, (m*Fr - C,,,R*R*’
(10)
+ J*‘), (C,,R*
The superscript T denotes transpose; the primes denote differentiation with respect to z*; We is the Weber number; N is the convergence parameter; C,, is the pressure coefficient; Fr is the Froude number; T is a nondimensional time; the starred variables are dimensionless; and R, and V, denote the nozzle mean radius and total velocity at the nozzle exit, respectively. Note that U, = v, cos 0,
v, = V, sin 0,
(12)
where U, and u, denote the axial and radial velocity
128
Appl.
Math.
Modelling,
1991,
Vol.
15, March
- J*‘IR*‘)IT
(11)
components at the nozzle exit, respectively, and 8, is the angle that the tangent to the liquid curtain mean radius forms with the z-axis at the nozzle exit, that is, at2
= 0.
The boundary
m, = ,&>bo
conditions UC>=
at the nozzle exit are
V, cos 8,
0,
=
V, sin
e,, (13)
and the nondimensional boundary nozzle exit can be written as
conditions
at the
Unsteady
liquid curtains:
J. I. Ramos
T
(14)
where h,lR, is the liquid curtain thickness-to-radius ratio at the nozzle exit. Equation (8) represents a system of partial differential equations. Note that the left-hand side of equation (8) is a convection operator that involves only firstorder derivatives with respect to r and z*, whereas the right-hand side of the equations for M* and N* involve R*“, that is, a2R*ldz*‘. The values of pi and ,D<,and initial conditions must be specified in order to solve equation (8). In this paper it is assumed that the gases surrounding the liquid curtain are dynamically passive so that pt, is constant, that is, the liquid curtain falls in a surrounding medium of infinite extent. The value of pi depends on the gas absorption by the liquid curtain, gas mass injection in the volume enclosed by the curtain, mass flow rate fluctuations at the nozzle exit, and so on. At high Peclet numbers the gas absorption by the liquid curtain is small, and in the absence of fluctuations at the nozzle exit and if the liquid curtain is impermeable, pi can be related to the mass of gases enclosed by the liquid curtain as follows. If the gases enclosed by the liquid curtain are ideal, pi Vi = miR, T;
(15)
where Vi, m;, and T, denote the volume, mass, and temperature of the gases enclosed by the liquid curtain, respectively, and R, is the gas constant. In the absence of both mass injection into and gas absorption by the inner surface, that is, r = R;, of the liquid curtain, mj is constant. Furthermore, if T, is also assumed to be constant, then equation (15) can be written as pi = miRn TiI
vRf dz
(16)
where L denotes the liquid curtain convergence length, that is, the axial distance measured from the nozzle exit at which RJL, t) = 0. This condition determines L = L(t), which is not known a priori. Note that Ri = R - b/2
R, = R f b/2
Equation (17) can be nondimensionalized R,+ = R* - b*/2
R,* = R* + b*/2
(17) as (18)
where RT = RiIR,
b* = blR,
R: = RJR,
(19)
Furthermore, according to the nondimensionalization of m = pRb introduced in equation (7), b* = m*I(R*N)
= m*21(RfN)
integrodifferential equations whose downstream boundary, that is, z, = L, is not known. In this paper we will assume that C,,,,, that is, (pi - p,) is a specified function of time, so that equation (8) represents a forced distributed parameter system; therefore equation (16) will not be used. The reasons for this seemingly academic study is to determine the dynamic response of liquid curtains to imposed pressure fluctuations. Future studies will consider equation (16), that is, cases in which there are velocity or mass flow rate fluctuations at the nozzle exit and unsteady mass injection into the volume enclosed by the liquid curtain. Furthermore, analytical studies for long liquid curtains, that is, L >> R,,, have shown that for 0,, = O”, C,,,, > 1, the liquid curtain does not converge, that is, R(z, I) # 0. These analytical studies have also shown that if 0,, = 0” and C,,,, = 1, then R(z, t) = R,,, that is, a cylindrical annular jet is obtained. In this paper we will also determine whether the critical value of C,,,, = 1 for steady-state, long liquid curtains can be exceeded while ensuring that the convergence length is finite, that is, L < a. Furthermore, if this condition is satisfied and the convergence length fluctuates in time, the mass absorption rate (which is ignored in this paper) may be enhanced. Numerical
method
As was discussed in the previous section, equations (l)-(3) and (5) are a boundary value problem whose downstream boundary, that is, L” = LIR,,, is an unknown function of time that must be determined so that R*(L*, 7) = 0. Since the location of the downstream boundary is an unknown function of time, it is convenient to introduce the mapping (7, z*) -
(5717)
(21)
77= z*lL*(r)
(22)
where t=r
so that 0 I n 2 1, that is, in (5, q)-coordinates the convergence length corresponds to the fixed boundary given by 77 = 1, and the nozzle exit corresponds to n = 0. Introducing equations (21) and (22) into equation (S), we have
!$ + W*$ = G*(I/*)
(23)
where
(24)
(20)
Equation (16) and the definition of C,, in equation (7) indicate that equation (8) represents a system of
c~F*Ic?U* is a 4 unit matrix.
x
4 Jacobian matrix, and I is a 4 x 4
Appl. Math. Modelling,
1991, Vol. 15, March
129
Unsteady
liquid
J. I. Ramos
curtains:
Equation (23) can be discretized by using backward differences in time and upwind differences in space as
= A
+ Vi””
(25)
where A[ is the time step; AT is the grid spacing; the superscript n refers to time, for example, 5” = nA& and the subscript i denotes the axial location of the ith grid node. Equation (25) employs upwind differences for the convection terms and is O(A& A+accurate. Note that the algebraic multiplicity of the eigenvalue
tiplicity, that is, the number of linearly independent eigenvectors, is three. Note also that 0 i 775 1 and u* > dL*ldT, so that h > 0. This is the reason why upwind differences were employed to discretize the convection terms in equation (23). The term G* was di;;;tized by means of second-order accurate forEqiation (25) is a block bidiagonal system; the block dimension is 4 x 4. This system can be solved by forward substitution once L* and dL*ldr are known. To calculate L*, we consider the continuity equation, that is, the first equation of equation (8), (27)
(26) of the matrix H* is four, whereas its geometric
mul-
Equation (27) can be integrated from z” = 0 to 2* = L*(T) to yield, after application of Leibnitz’s rule,
(28) because m*(O, T)u*(O, 7) = M*(O, T) = N$
0
Fr
(2%
where equation (14) has been used. Substitution of equations (21) and (22) into equation (28) yields dL* -=&[$(L*[m*dq) dt
which can be written in finite difference L*“+l
=
(30)
+M*(l,t)-N$Fr]
- im*‘fdq]
m*“+‘(l,[)
form as L”” + [M*“+‘(l,t)
- N$Fr]
A,$}/(m*““(l,f)
0
0
- irn*‘l+ldq) 0
where the time derivatives have been discretized by means of backward differences. (31) Equation (25) was solved by means of an iterative block-implicit method as follows. Steady-state calculations were first performed for fixed values of b,,lR,,, C,,,,, N, I%, and Fr to determine the corresponding steady state values of r/f and L*. These values were used as initial guesses for UT”+’ and L*” and employed to solve equation (25) with dL*ldT = 0. The resulting values from the solution of equation (25) were UTkf’ and were used to determine Lfh+ ’ from equation (31). If
5
[(ui*k+’
U32]1/2
-
5
10
-4
(32)
i= I
where k denotes the kth interation and NP is the number of grid points used in the calculations, convergence was achieved within the time step; otherwise, an iterative procedure was used to solve equations (25) and (31). When performing steady-state calculations, that is, C,,, # Cpn(~), the following criterion was used to determine the steady state: jyi*n+’
130
AE>I -
UT”
2
“Z<
Appl. Math. Modelling,
,o_4
(33)
1991, Vol. 15, March
Note that ((ji*ll+’ - Ui*“)2 = (I/i*n+l - ui*“)(ui*n+’ - UT”)T
Presentation
(34)
of results
The transient calculations presented in this paper were performed with 401 grid points, that is, AT = 2.5 x 10e3, and At = 10P2. Calculations were also performed with smaller values of AT and At to determine that the results are independent of the time step and grid spacing. Steady-state calculations were also performed for C,, = 0 with the same values of AT and
Unsteady liquid curtains: J. I. Ramos
At to determine
the steady-state
convergence
where C,,, and w are the amplitude and angular frequency of the imposed pressure fluctuations, respectively. The dynamic response of liquid curtains to step and ramp pressure coefficients has also been studied by Ramos and Pitchumani’by means of nonadaptive finite difference methods. By introducing the nondimensional variables of equation (7), equation (35) can be written as
length
L&.
Calculations were performed to determine the dynamic response of liquid curtains subject to step, ramp, and sinusoidal changes in the pressure coefficient C,,, using the domain-adaptive technique presented in this paper as well as a fixed finite difference method. However, only the response of the liquid curtain to sinusoidal variations in the pressure coefficient will be considered in this paper, that is, C,, = C,,, sin (wf) Table 1.
C,,, = C,,,, sin (2~ St Fr 7) where St = wRJ(27rVJ
(35) Values of the parameters
Figure
Fr
N
2 3 4 5 6 7 8 9 10
3 3 3 3 3 Variable 5 3 5
625 625 625 625 625 625 Variable 625 625
(36)
is the Strouhal number.
used in the calculations &JR,
00 0 0 0 0 0 0 0 Variable 0
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Variable
c JJna 1 1 1 1 Variable 0.5 0.5 0.1 0.5
St 11120 1160 1I30 1115 1115 1130 1I30 1130 1I30
CPn
1x
--L ‘L,,
IO
I,
20
Figure 2.
Convergence
length as a function of time
Figure 4.
Convergence
length as a function of time
Figure 3.
Convergence
length as a function of time
Figure 5.
Convergence
length as a function of time
Appl.
Math. Modelling,
1991, Vol. 15, March
131
Unsteady liquid curtains: J. I. Ramos
Equations (8)-(1 I), (14), and (36) indicate that the dynamic response of liquid curtains subject to sinusoidal pressure fluctuations depends on Fr, N, $,, b,lR,, c pna, and St. The values of the parameters used in the calculations presented in this paper are shown in Table 1. Figures 2-5 show the convergence length normalized by the steady-state convergence length L.&, that is, the convergence length corresponding to St = 0, as a function of time and of the Strouhal number for a fixed value of C,,,. Figures 2-5 show that the liquid curtain does not respond instantaneously to the imposed pressure fluctuation and that there is a lag between L*(T) and C,,,(T). This lag is due to the inertia of the liquid curtain and depends on the Strouhal number. The larger the Strouha1 number, that is, the larger the frequency of the imposed pressure fluctuations, the smaller the lag time. For times larger than the lag time the convergence length responds rapidly to the imposed pressure fluctuations and exhibits a periodic behavior whose amplitude and frequency decrease and increase, respectively, as the Strouhal number is increased. The periodic character of the convergence length is to be expected, since only inviscid liquids are considered in this paper and there is no damping. Figures 2-5 also indicate that after the initial lag time, L* decreases as C,, increases, and L* increases as C,,, decreases. This is again due to the inertia of the liquid curtain. Furthermore, the amplitude of the oscillations in the convergence length increases as the Strouhal number is decreased; for St = 11120, L&,, is about 4 L& (Figure 2), whereas for St = l/15, L&, is about 1.5 L,*,. Analytical solutions of the steady-state equations governing the fluid dynamics of liquid curtains have shown that for 0, = 0” the liquid curtain converges for C,,, < 1, that is, L * is finite. However, L* is infinite for 8, = 0” and C,, 2 1. These analytical studies are based on a small curvature-small slope approximation and indicate that R* = 1, that is, a cylindrical annular jet is obtained, for 19,,= 0” and C,,, = 1. The results presented in Figures 2 and 5 correspond to Cpnr, = 1
4,
(
l.O-
z.o----
cpna’
3
3.0
-___-
3.5__-__
I
and indicate that for C,,, = 1 the liquid curtain converges even though a steady-state liquid curtain would not converge for C,,, = C,,,, = 1. The finite convergence length observed in Figures 2-5 indicates that the liquid curtain feels a pressure coefficient lower than C,,, due to the liquid inertia and lag time. Figures 2-5 also indicate that the higher the frequency of the imposed pressure fluctuations, the smaller the amplitude of the convergence length oscillations. In the limit St + 00 the amplitude of the convergence length oscillations tends to zero. Figure 6 shows the convergence length as a function of the amplitude of the imposed pressure fluctuations for a fixed value of the Strouhal number. This figure clearly illustrates that the frequency of the convergence length oscillations is independent of the Strouhal number, whereas the amplitude of the oscillations increases as CD,, is increased. This behavior is expected because of the fact that there is no damping. Figure 6 also shows that the amplitude of the imposed pressure fluctuations could be as high as the critical value of unity corresponding to steady-state liquid curtains. Furthermore, Figures 2-6 indicate that the value of C pnu could be further increased while still obtaining finite convergence lengths if the Strouhal number were increased. The results shown in Figures 2-6 correspond to fixed values of Fr, N, 8,,, and b,lR,. In Figures 7-10 the effects of these parameters on the liquid curtain response are examined for fixed values of C,,,,,,and St. Figure 7 illustrates the effects of the Froude number on the convergence length of the liquid curtain. As is indicated in equation (36), an increase in the Froude number implies an increase in the nondimensional frequency of the imposed pressure fluctuations. As a consequence, the frequency and amplitude of the convergence length oscillations increase and decrease, respectively, as the Froude number is increased. Note that the steady-state convergence length, that is, the convergence length corresponding to C,, = 0, increases as the Froude number is increased. Furthermore, the results shown in Figure 7 indicate that liquid
-’
9 *3
2
I
0 0 Figure 6.
132
IO Convergence
i
length as a function of time
Appl. Math. Modelling,
1991, Vol. 15, March
Figure 7. Convergence Froude numbers
length as a function of time for different
Unsteady liquid curtains: J. I. Ramos I.5
I
.D i lA 0.5
Figure 8. Convergence length as a function of time for different values of the convergence parameter
Figure 9. Convergence length as a function of time for different values of the nozzle exit angle
curtains with low Froude numbers are very sensitive to the imposed pressure oscillations, whereas curtains with high Froude numbers can withstand higher amplitudes of the imposed pressure oscillations. Figure 8 shows the convergence length as a function of the convergence parameter and clearly illustrates the effects of the liquid curtain inertia on the liquid curtain response. As is indicated in equation (7), the convergence number is the ratio of the inertial forces to surface tension for a fixed value of the Froude number, that is, the liquid curtain inertia increases in relation to the surface tension as the convergence parameter is increased. Therefore as the convergence parameter is increased, the amplitude of the convergence length oscillations decreases, and in the limit N+ m the amplitude of these oscillations tends to zero. Note also that the lag time between the imposed pressure oscillations and the oscillations of the convergence length increases as N is increased, but the frequency of the convergence length oscillations is independent of the convergence parameter because both St and Fr were fixed and there is no damping.
Figure 10. Convergence length as a function of time for several thickness-to-radius ratios
Figure 9 shows the convergence length as a function of the nozzle exit angle. Negative values of 0, correspond to flows directed towards the symmetry axis. Figure 9 indicates that the amplitude of the convergence length oscillations increases as 0, is increased. This behavior is to be expected, since positive values of 0, result in longer liquid curtains that are subjected to the imposed pressure oscillations over a larger surface area. Furthermore, the radius of the liquid curtain increases, whereas its thickness decreases, as 0, is increased. Note that the results shown in Figure 9 indicate that the frequency of the convergence length oscillations is independent of the nozzle exit angle. Figure IOpresents the convergence length as a function of nozzle gap width and indicates that the lag time increases as b,lR,, is increased, that is, as the liquid curtain inertia is increased. The amplitude of the convergence length oscillations increases as b,,lR,, is decreased, whereas their frequency is independent of the liquid curtain thickness-to-radius ratio at the nozzle exit. Conclusions The asymptotic equations governing the dynamics of liquid curtains subject to imposed pressure fluctuations have been solved numerically by means of a domainadaptive, iterative block-implicit method. The numerical method uses backward differences in time and upwind differences for the convection terms and results in a block bidiagonal system that can be solved by forward substitution. The main advantage of the domain-adaptive method presented in this paper is that the calculations are performed in a computational domain with fixed boundaries. However, an ordinary differential equation must be derived to determine the liquid curtain convergence length as a function of time. In the calculations presented in this paper that ordinary differential equation was derived from the continuity equation and ensures global mass conservation.
Appl.
Math. Modelling,
1991, Vol. 15, March
133
Unsteady
liquid curtains:
J. I. Ramos
Calculations were performed to determine the effects of the amplitude and frequency of the imposed pressure fluctuations, Froude number, nozzle angle, convergence parameter, and nozzle exit thickness-toradius ratio on the convergence length of the liquid curtain. These calculations show that the amplitude of the convergence length decreases as the frequency of the imposed pressure fluctuations, Froude number, convergence parameter, and nozzle exit thickness-toradius ratio are increased and increase as the nozzle exit angle and amplitude of the imposed pressure fluctuations are increased. The frequency of the convergence length oscillations increases as the Froude number and the frequency of the imposed pressure fluctuations are increased, but they are independent of the nozzle exit angle, convergence parameter, amplitude of the imposed pressure fluctuations, and nozzle exit thickness-to-radius ratio. It has also been shown that the critical value of the pressure coefficient determined from analytical solutions of the steady-state equations governing the fluid dynamics of liquid curtains can be exceeded if the curtain is subjected to pressure oscillations of sufficiently high frequency. This implies that stable liquid curtains can be obtained if the imposed pressure oscillations have high frequency. The calculations also show that the initial response of the liquid curtain is slow for high values of the nozzle exit thickness-to-radius ratio and convergence parameter and for small values of the Froude number and frequency of the imposed pressure oscillations.
m
mass per radian and per unit length of the curtain m = pRb dimensionless axial momentum per unit length and per radian convergence parameter, N = We/F? nondimensional radial momentum per unit length and per radian number of grid points hydrostatic pressure radial coordinate mean radius of curtain
m” N N* NP P k RT St
RT = m*R*
Strouhal number, St = wR,l(2rrV,) time axial component of the velocity of the liquid curtain four-dimensional vector defined in equation (9) radial component of the velocity of the liquid curtain velocity of liquid curtain, V = (u* + v*)“* Weber number, We = pV;R,,I2u axial coordinate
t U
u*
V
V
We 2
Greek symbols step size nondimensional axial coordinate defined in equation (22) eigenvalue angle between the tangent to the liquid curtain mean radius and the z-axis nondimensional time defined in equation (22) density of the liquid curtain surface tension of the liquid dimensionless time angular frequency of the imposed pressure fluctuations
Acknowledgments This work was supported by the Oftice of Basic Energy Sciences, U.S. Department of Energy, under Grant No. DE-FG02-86ER13.597 with Dr. Oscar P. Manley as technical monitor. This support is deeply appreciated. The author also appreciates the support provided by CRAY Research, Inc., through grants from the 1988 and 1989 CRAY Research and Development Grant Program, and by the Pittsburgh Supercomputing Center.
Nomenclature b
C Pn c PflU Fr F*( U*)
g G( V*) H*
Z J’ L
134
max 0 ss I
liquid curtain thickness pressure coefficient defined in equation (7) amplitude of oscillation of C,,, Froude number, Fr = V2/gRo four-dimensional vector defined by equation (IO) gravitational acceleration four-dimensional vector defined by equation (11) 4 x 4 matrix defined in equation (24) unit matrix term defined in equation (4) convergence length of the liquid curtain
Appl. Math. Modelling,
Subscripts e i
1991, Vol. 15, March
* T
outer surface of the liquid curtain inner surface of the liquid curtain maximum nozzle exit steady state corresponding to C,, = 0 Superscripts
differentiation coordinate dimensionless transpose
with respect to the axial quantity
References 1 2 3
Roidt, R. M. and Shapiro, Z. M. Liquid curtain reactor. Rept. No. 85M981, Westinghouse R&D Center, Pittsburgh, Pa., 1985 Ramos, J. I. Liquid curtains. I: Fluid mechanics. Chrm. Engrg. Sci. 1988, 43, 3171-3184 Ramos, J. I. Analytical, asymptotic and numerical studies of liquid curtains, and comparisons with experimental data. Appl. Math. Modding 1989, 14, 170-183
Unsteady liquid curtains: J. I. Ramos the Sixth Symposium on Energy Engineering Sciences: Flow und Transport in Continuu, U.S. Dept. of Energy, Argonne National Laboratory, Argonne, Illinois, May 1988, pp. 18-25
Ramos, J. 1. and Pitchumani, R. An analysis of laminar boundary layers on liquid curtains. J. Appl. Math. Phys. (ZAMP) 1989, 40, 72 l-739
Ramos, J. I. and Pitchumani, R. Gas absorption by liquid curtains. Rept. CO/88/3, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh, Pa., July 1988 Ramos, J. I. Annular liquid jets in zero-gravity. Appl. Muth. ModeUing 1990, 14, 630-640 Chigier, N., Ramos, J. I. and Kihm, K. Experimental and theoretical studies of vertical annular liquid jets. Proceedings of
8
9
Ramos, J. 1. Liquid membranes: Formulation and steady state analysis. Rept. CO/89/4, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh, Pa., February 1989 Ramos, J. I. and Pitchumani, R. Unsteady response of liquid curtains to time-dependent pressure oscillations. Rept. CO18814, Dept. of Mechanical Engineering, Carnegie-Mellon Univ., Pittsburgh. Pa., September 1988
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