PROPERTIES OF V AND V CLASSES OF SOLUTION TO THE THREE-DIMENSIONAL INCOMPRESSIBLE NAVIERSTOKES EQUATIONS G. Nugroho, A. M. S. Ali, and Z. A. Abdul Karim Department of Mechanical Engineering, Universiti Teknologi Petronas, Bandar Seri Iskandar 31750, Tronoh, Perak, Malaysia Email:
[email protected] Received 19 November 2009; accepted 25 March 2009
ABSTRACT The solution in the special classes of V and V of Navier-Stokes equations is investigated in this work. Analysis is taken using the vorticity equations rather than the original Navier Stokes equations based on the work of Chae and Choe (1999). Derivations show that the corresponding problem can be transformed to the class of linear parabolic and elliptic equations. Analysis is performed on L , L2 and Lp theory of solutions to linear problems. Results show that the corresponding problems admit unique and regular solutions. These properties are also confirmed in the derivation in order to find exact solutions. Thus several validation cases are also presented for laminar and turbulent cases including combustion. Keywords: Continuity equation, the Navier-Stokes equations, vorticity equations, potential function, partial differential equations, analytical solution.
1 INTRODUCTION Despite the concentrated research on Navier Stokes equations, their universal solution is not achieved. The full solution of the three-dimensional Navier-Stokes equations remains one of the open problems in mathematical physics. The existence and smoothness theorem is widely applied to the mathematical analysis of Navier-Stokes equations as described in many literatures. For example, nonstationary Navier-Stokes equations in the entire threedimensional space are observed (Panel and Pokornyi, 2004). Some criteria on certain components of velocity gradient are given to ensure global smoothness in time (Zhou 2006). However, global in time continuation still remains an unsolved problem in mathematical fluid mechanics (Smith 2006). The nonstationary Navier-Stokes equations in 0,T are also considered (Suzuki 2008), and the regularity of suitable weak solutions is proven for large x . It is also mentioned that their result also holds near the boundary. More general regularity with simplified problem is investigated (Cheng 2002), which states that u3 satisfies either u3 L 2 0, T or u3 Lp 0, T ; Lq 3 with 1 p 3 2q 1 2 and q 3 for some T 0 then u is regular on 0,T . Similar investigation is performed for thin three-dimensional flows Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
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(Montgomery and Smith 1999). It is concluded that the solution of the Navier-Stokes equations on the corresponding domain with periodic boundary conditions has global regularity, as long as there is control on the size of the initial data and the forcing term. Also, Navier-Stokes equations are modified in a lengthy work (Seregin 1999) to find the interior regularity and to ensure the uniqueness of the solutions. However, it is possible to generate the existence theorem from explicit solutions like numerical methods (Heywood et al. 1999) to provide, by strict solution, a rigorous a posteriori analysis of the existence of the steady solutions. It is clear that although it is promising to overcome the problem of nonlinear differential equations by finding classes of exact solution (Galaktionov and Svirshchevskii 2007), it is important to give the foundations of the analytical solutions to explore their global properties, like one might find possible local singularity for this particular class of the solution (Kuczma et al, 1990). Therefore, this work is conducted as to support the analysis of the analytical solution proposed previously by the same authors (Nugroho et al. 2009). Analysis is carried out in vorticity equations rather than Navier-Stokes equations by considering that the solutions will fulfill some certain conditions that satisfy the Navier-Stokes equations (Chae and Choe 1999). In this work, a potential function is proposed to form the special classes of solution.
2 PROBLEM FORMULATION Navier Stokes equations in Cartesian form for incompressible flows are written as, V 1 V V p 2V t
(1a)
is static pressure, is fluid density, is kinematic viscosity and x , y , z . The solution of the system of equations describes the three velocity components in the three spatial directions, i.e., V u, v, w , where u u x, y, z, t , v v x, y, z, t and w w x, y, z, t . However, it is more reliable to study the fluid motion using vorticity (Kozono and Yatsu 2004). By taking curl operation to the Navier-Stokes equations, the following vorticity equations are obtained, where
p
V .V 2 t
(1b)
with V and x , y , z . The vorticity components are also distributed in the three dimensions, x x x, y, z, t , y y x, y, z, t and z z x, y, z, t . Note that the pressure term in (1a) is vanished by the curl procedure and that the solutions of (1) shall satisfy the continuity equation for incompressible flow, V 0
(2)
It is supposed that there exists a potential function such that the velocity vector can be expressed as, V Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
(3a)
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Therefore, the vorticity can also be expressed using the potential function as,
(3b)
Thus, the velocity and vorticity can be defined explicitly as u y z , x xy yy zz xz
(4a)
v z x , y yz zz xx xy
(4b)
w x y , z xz xx yy yz
(4c)
Substitute equation (4) into the vorticity equations (1b) will yield a system of the following equations, In x direction;
xy yy zz xz t
x y
y z
xy yy zz xz z
xz xx yy yz
y z
z
2 xy yy zz xz z
xy yy zz xz
x
xy yy zz xz
z x
y z x
xy yy zz xz
y
yz zz xx xy
2 xy yy zz xz 2 xy yy zz xz x 2
y z
y
y 2
2
(5a) In y direction;
yz zz xx xy t
x y
y z
yz zz xx xy z
xz xx yy yz
2 yz zz xx xy z
z x z
yz zz xx xy x
z x
yz zz xx xy
y
xy yy zz xz zx x yz zz xx xy zy x
2 yz zz xx xy x 2
2 yz zz xx xy y 2
2
(5b) In z direction;
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xz xx yy yz t
x y
V
y z
xz xx yy yz z
xz xx yy yz
Properties of
x y z
2 xz xx yy yz
and
V
xz xx yy yz
x
xy yy zz xz
z
x
x y x
xz xx yy yz
y
yz zz xx xy
2 xz xx yy yz 2 xz xx yy yz x 2
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x y
y
y 2
z 2
(5c) By taking the sum of equation (5), the following equation is produced, i 2 t
(6)
where 2 xy xz yz , 1 y z , 2 z x , 3 x y , and where some terms are vanished because of the continuity. Therefore, the corresponding problem falls into a category of simpler parabolic differential equations. Now a more general solution to the three-dimensional Navier-Stokes equations will be considered using the function, V
(7)
Similar expression to (6) can be obtained by the same procedure as above where the following terms are added to the right hand side of (5), for x direction :
xy yy zz xz
x y z x
yz zz xx xy
x y z y
xz xx yy yz
x y z
z
(8a) for y direction :
xy yy zz xz
y z x x
yz zz xx xy
y z x y
xz xx yy yz
y z x
z
(8b) for z direction :
xy yy zz xz
z x y x
yz zz xx xy
z x y y
xz xx yy yz
z x y
z
(8c) Thus, according to the differentiation rule, by assuming that equation (8) is equal to,
xxy xyy xzz xxz x y z yyz yzz xxy xyy x y z xzz xxz yyz yzz x y z (9)
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for all direction, so zero result will be observed. Then, because of (7), components of equation (6) are redefined as, 2 xy xz yz , 1 x y z , 2 y z x and
3 z x y . Equation (6) can be transformed further by taking x ln 1 , y ln 2 , z ln 3 ,
to give,
2 2j 2J t j 2j
(10)
which becomes a linear parabolic equation with respect to i and has well defined solutions as proved in the next section. With this result, can be rewritten in terms of the potential function, as, 2 xy xz yz f j f j
(11)
and may be investigated by nonlinear analysis which depends on the solution of j . However, it is interesting to note that trivial form of linear differential equations can be generated from equation (11). Equation (11) is redefined by transforming back f j to their original form in x, y, z, t , 2 xy xz yz f x, y, z, t
(12)
Equation (12) also falls into the category of linear elliptic differential equations.
3 THEORY OF SOLUTIONS This section concentrates on the existence and uniqueness of the classical, weak and strong solutions. The analysis is based on the maximum principle of the linear parabolic equation in (10), which is implemented to determine L norm estimate and comparison principle (Zhuoqun et al. 2006). Theorem 1: Let j 0 and bounded in , C 2 satisfies equation (10). Then,
sup j , t sup j , t
Proof: Consider the existence of point 0j , t 0 at such that,
0j , t 0 sup j , t
0
(13a)
The maximum principle asserts the following condition,
0j , t 0 t
0, , t 0 , , t 0 0 j
0
2
0 j
0
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(13b)
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Then equation (10) will result in,
0j , t 0 t
, t , t 0 2 j
0 j
0
2 j
2
0 j
0
(13c)
and equation (13a) is valid. Let g e t with 0 and g C 2 , substituting into equation (10) will result in, L
g e t 0 t
Then, for any constant 0, L f L Lg 0
(13d)
According to the above equation and (13a), the following inequality is produced,
sup j , t g j , t sup j , t g j , t
(13e)
Let 0 , thus theorem 1 is proved. Therefore, an additional result can also be concluded as below Theorem 2: Suppose that j 0 and bounded in , 1 , 2 C 2 satisfies L1 L2 in with
1 0j , t 0 2 0j , t 0 at . Then 1 , t 2 , t in .
By theorem 1, the initial-boundary value 0j , t 0 can be chosen to ensure the a priori bound for solutions of (10), theorem 2 also ensures that 3 1 2 0 . Hence the existence and uniqueness of classical solutions for (10) are proved. The L2 theory of equation (10) can be stated in the following, Theorem 3: For L2 , the initial-boundary value problem of (10) admits at most one weak solution. Proof: (Uniqueness). Let 1 and 2 be weak solutions of initial-boundary value problem (10), if 3 1 2 W21,1 and fulfill,
T
3 2j3 2j 3 . d dt 0 t
Choosing 3 and the maximum principle reveals
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T
3 3 2j 33d dt t
2 j
2
3 d dt 0
(14b)
T
Poincare inequality is then implemented to obtain
d dt 0 2 2 j 3
(14c)
T
Therefore, 3 0 and 1 2 in which ensure the uniqueness of weak solutions. (Existence). Considering,
t e 2 j
s
d dt
T
2 j
T
2
e
s
s T
d dt ,
(14d)
The right hand side is equal to,
2 j
2
T
e
s
2 2 d dt 2j d e s dt 2j e s d dt T T
(14e)
and (14d) will change as follows,
t e 2 j
s
d dt
2 j
T
T
2
e
s
d dt
(14f)
Poincare inequality in the form,
e 2 2 j
s
d dt C
T
2 j
T
2
e
s
d dt
will take us to,
2 j
2
T
e
s
d dt
1 C
e 2 2 j
s
d dt
T
t
T
2 j
2 j
2 e s d dt
(14g) Therefore, there exist, W 1,1 3 2
t
T
2 j
2 j
2 e s d dt
(14h)
as weak solutions to the initial-boundary value problem of (10). The existence and uniqueness of solutions with intermediate regularity is based on the Lp theory as follows, Theorem 4: For Lp , the initial-boundary value problem of (10) admits a unique strong solution Wp2,1 Wp1,1 . Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
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Proof: Multiplying both side of equation (10) by p2 and integrating over and T as,
T
p 2 p 2 2j d dt t
2 j
p 2
2d dt
(15a)
T
Integrating by parts over spatial coordinate to yield, 1 p
T
p
1 d dt t p
2j
p
d dt
4 p 1 p2
T
T
2
2j
p 1 2 d dt
(15b)
Multiplying by e s as in theorem 3, and by similar procedure the following result is obtained, W 2,1 3 W 1,1 3 p
p
1 p
T
t
p
d dt
1 p
p
2j d dt
T
4 p 1 p2
T
2j
p 1 2
2
d dt
(15c) Thus, suppose that 1 , 2 Wp2,1 Wp1,1 are strong solutions, and based on the estimate of the maximum principle and Poincare inequality, the following is obtained, 4 p 1 p
2
2 j
T
p 2 1 d dt 0
(15d)
Setting 3 1 2 0 then 1 2 and uniqueness is also proved. The elliptic equation (12) is easier to be analysed since f , t is proved to be bounded. Here the existence and uniqueness of the regular solutions of (12) will be demonstrated. Theorem 5: Let f be bounded, the boundary value problem (12) admits a unique classical solution. The proof is similar to that of theorem 1. Theorem 6: For any f L2 and bounded, the boundary value problem (12) admits at most one solution. Proof: Multiplying (12) by , then there exists a unique H 1 such that,
. d f d ,
H 1
This shows the existence of the weak solutions of boundary value problem of (12). Theorem 7: For any f Lp , equation (12) admits a unique strong solution W 2, p W 1, p .
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Proof: The proof here is different than that of theorem 4 since it is guaranteed that f Lp is bounded. Multiplying (12) by p 2 and the relation over is,
p 2
2 d
f
p
d
(17a)
Integrating, 4 p 1 p2
2
p 1 p 2 d f d
(17b)
By Poincare inequality, Holder inequality and Young inequality, equation (17b) will transform to, 4 p 1 Cp
2
f
p
d
Lp
f
p 1
d
p 1 LP
p
d
1 p 1
(17c)
f
p
d
where C is constant in Poincare inequality and is constant in Young inequality, and the above result leads to, W 2, p C f
(17d)
Lp
Let 1 , 2 W 2, p W 1, p and set 3 1 2 then the estimate (17d) will ensure the uniqueness of strong solutions. This proves the theorem. Hence, the initial-boundary value problem of (10) and (12) proves have generalised unique solution. If i is held, then i will admit general classical solution and will have global regularity for weak solution in L2 and strong solution in Lp with the assumption of regular boundary.
4 TWO EXAMPLES OF EXPLICIT ANALYTICAL SOLUTIONS AND THEIR PROPERTIES Now, consider a potential function , so that the velocity components are the derivatives of the function and can be expressed in vectorial form as, V
and
V
(18a)
with x , y , z . The spatial coordinates are transformed into a single coordinate through the following transform function, kx ly mz ct Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
(18b)
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where k , l , m and c are constants. The velocity components in equation (18a) can be rewritten including the new coordinate. Then, substituting to the Vorticity equations (1b) and add them all to give, 2
2 3 3 4 a 3 b d e 2 3 4
(19)
where the coefficients appear in (19) are also constants. Note that either vectorial velocity V or V will produce the same equation (19) with different coefficients. Theorem 8: The initial boundary value problem of (18a), (18b) and (19) have unique point values. Proof: In order to prove theorem 8, the following statement is considered firstly, Lemma 1: Equation (19) can be expanded into a second order polynomial differential equation and has solution as,
g1 ln 1 e g2 g3
where g1 , g2 and g3 are constants. Proof of Lemma 1: Implementing Q to yield, 2
a
Q 2Q 2Q 3Q bQ d e 2 2 3
Take
Q R
(20a)
then equation (20a) will transform into, 2
a
R R R 2 R bQ dR e evR 2 Q Q Q Q
(20b)
By expanding R into a polynomial form in Q , it becomes reasonable that the resulting problem falls in the class of Riccati equation as, R
Q f1Q2 f 2Q f3
(20c)
where f1 , f 2 and f3 are constants obtained from the substitution of (20c) into (20b). The substitution is,
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2af1Q af 2 2bf1Q 2 bf 2Q df1Q 2 df 2Q df 3 4evf12Q 2 4evf1 f 2Q evf 22
2evf12Q 2 2evf1 f 2Q 2evf1 f 3
(20d)
The coefficients are then grouped and rearranged to give, 6evf12 (d 2b) f1 0 6evf1 f 2 df 2 2af1 bf 2 0 2evf1 f 3 evf 22
(20e)
df 3 af 2 0
and f1 , f 2 and f3 are obtained. Obtaining a solution of (20c) then the expression for potential function is written as,
g1 ln 1 e g2 g3
(20f)
where g1 , g2 and g3 are constants. This proves lemma 1. However, it is interesting to discuss the uniqueness property by generating more solutions. The other example is shown in the statement below, Lemma 2: Let R a bQ , equation (19) is then transformed into Riccati equation and its solution is represented as, 1 g 4 ln r
e
k d d g 5
where g 4 and g5 are constants. Proof of Lemma 2: Take R a bQ in (20a) such that the equation will transform into, 2
1 2 R d R e 3R R b 2 b2 b 3
(21a)
Let b d and integrate (21a) once to give, e 2 R 1 R R g b 2 d
(21b)
Integrate once more and rearrange, the above equation will become, e R 1 2 R g h b 2 d
Let R 2d
S S
(21c)
then equation (21c) transforms to a second order linear differential equation,
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V
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Classes of Solution
n S 0
(21d)
Here the interesting method proposed by Bougoffa (2009) is extended to find the closed form solution of (21d). Multiplying (21d) by a function q to obtain,
qS q S nqS 0
(21e)
Multiply once more by an arbitrary function r and let q r nqr then (21e) can be redefined as, 1 1 r q G G q G 0 r r
(21f)
where G rS . Since r is an arbitrary then the condition q const can be imposed. r Therefore, equation (21f) will become, 1
2G G k 0 2
(21g)
and it has solution written as below, G e
k d
Thus, S
d
(21h)
e r 1
1 g 4 ln r
e
k d d and substituting into R a bQ 2d S , then is represented as, S
k d d g 5
(21i)
where g 4 and g5 are constants. This proves lemma 2. Performing velocity vector through (18a) and reverse transformation (18b), by implementing initial and boundary values in (20f) and (21i), the solution constants can be obtained. Note that the resulting velocity vector must be the same for (20f) and (20i) to ensure uniqueness. Substitute arbitrary values t * , y * and z * in the solution, then a unique value for x * is found. The process then can be repeated by induction to find any other unique points. This completes the proof of theorem 8. However, by applying Q
1 S , equation (20c) will transform to a second order linear f1 S
equation, together with (21g) the uniqueness property can also be ensured by abstract analysis. Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
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Then, by performing (18a) and substituting into the Navier-Stokes equations (1a), the pressure relation is also obtained. However, it is possible that the initial and boundary value problem generate blow up solutions and need regularization procedure. The regularization procedure that is proposed here depends on finding an approximation expression for V . Let V be a blow up solution of the Navier-Stokes equations as t T , then the modified regularized solution can be written as, V*
V V
(22a)
where is a very small number. Integrate (22a) with respect to V , the expression for the modified solution is written as follows, V ln V
V
M N
1 V dV
V *
V * 1 V*
2
(22b)
dV *
2V V 1 . The 2 2 and N V V modified equation (22b) is finite as t T and the right hand side converges to V with the M controllable error, ln V 1 dV . V N V 2
where is an arbitrary small number, M
5 SIMULATIONS The analytical solution is validated and also presented in this paper. Following the method in the previous section, the explicit solutions for the proposed class of the respective potential function can be varied. In this section, the validation procedure started with laminar cases. The first validation case is the laminar free jet experiment of Symons and Labus (1971). A jet is the flow generated by a continuous source of momentum. The Reynolds number of a jet can be conveniently defined as Re
u0 D
, where u0 is jet velocity, D is jet diameter and is
kinematic viscosity of the fluid. The prescribed data is the normalised downstream velocity. Fig.1 shows a comparison between calculated centerline downstream velocity using the analytical solution and the measured velocity. As shown, the analytical solution could reproduce the decay of the measured downstream velocity with longitudinal distance from the nozzle. In the figure, both experimental data and analytical calculations are normalized. It is observed that the comparison for higher velocity (higher Re) is more accurate. It may be due to the characteristic of the solution itself. Analytical solutions are obtained through the simple coordinate transformation. By dimensional analysis, it is clear that contributions of viscous terms are weakened for higher Reynolds number as described below, U 1 2 U .U P U Re
with Wt , L , U u W and Re WL v .
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Figure 1: Decaying velocity along downstream direction produced by analytical solution (solid line) and the experimental data (points) (Symons and Labus 1971)
More comparison of the velocity profile is shown in figure 2. The transverse velocity is the one used for comparison here. The calculated values follow the same trend as the measured ones with high accuracy. However, for some points there is slight deviation which can be attributed to the vortex formation around the longitudinal axis immediately when the flow jets out of the nozzle exit.
Figure 2: Gaussian velocity along transversal distance produced by analytical solution (solid line) and the experimental data (points) (Symons and Labus 1971)
The other similar experiment used for validation is the laminar free jet of Eappen (1991). Figure 3 shows a comparison of the velocity profile calculated using the analytical solution predictions with the measured values. The inlet boundary condition is based on parabolic velocity profile to match the experimental set up. Different from the decay velocity, comparisons for transverse velocity profile show that calculation for higher velocity is less accurate than the other. This might happen due to the vortex formation of the flow as it leaves the nozzle. The vortex formation has a general tendency to produce and accumulate the eddies along the longitudinal direction. The process is happen through entrainment which draws the material outside into the jet.
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Figure 3: Velocity profile in transverse coordinate performed by analytical solution (solid line) and the experimental data (points) (Eappen 1991)
The third validation case is a water jet of diameter D = 10 cm and discharge velocity 10, 20 and 30 m/s (Abdel-Rahman et al. 1997). Figure 4 shows the measured radial profile of the normalized time-mean axial velocity at transversal locations for the turbulent round jet. It is seen that the streamwise velocity profile is similar to that of the laminar cases and can be well approximated by a half-Gaussian distribution.
Figure 4: Turbulent velocity profile along transversal direction produced by analytical solution (solid line) and the experimental data (points) (Abdel-Rahman, Chakroun and AlFahed 1997)
In figure 5, the calculated centerline velocity variation for the round turbulent jet is plotted against the measured values. The experimental results show clearly the existence of a potential core for about three diameters from the source, and the predicted variation confirms the experimental observation well. Previous experimental and analytical transverse velocity (Deo et al. 2007, Xu et al. 2008) shows similar trends to the case under discussion. In their work it could be seen that the transverse variation is similar, the data at different sections lie nicely onto one curve and can be well-approximated by the Gaussian distribution. Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
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Figure 5: Turbulent decay velocity along downstream direction produced by analytical solution (solid line) and the experimental data (points) (Abdel-Rahman, Chakroun and AlFahed 1997)
It is well known that turbulent flows are much more complicated than laminar flows, and thus some naïve prediction approaches will fail for turbulent flows even if they were successful for simple laminar flows. Therefore, the analytical solution needed to go through a second stage of validation against turbulent flow cases. The first turbulent flow case chosen for this validation stage is a boundary layer in atmospheric flow experiment of Farrel and Iyengar (1999). In this experiment, data were produced in a 1.7 m wide, 1.8 m high and 16 m long test section of the St. Anthony Falls Laboratory tunnel. The experimental technique was based on the use of quarter-elliptic, constant-wedge angle spires with height of 1.2 m and a castellated barrier wall to produce the necessary initial momentum defect in the boundary layer, followed by a fetch of roughness elements representative of the terrain under consideration. Fig. 6 gives a comparison of the calculated boundary layer velocity profile produced by the analytical solution and the measured profile from the experiment. The analytical results are in good agreement with the experimental data. Note that analytical solution described here is similar to the famous Blasius solution for boundary layer flows. Blasius solution for rectangular coordinate follows, 2 f ''' ff '' 0
(24)
where all parameters above are non dimensional. Equation (21) is a class of quasi linear differential equation and similar to (19b) for in its asymptotic limit and its solutions resemble previous solutions, thus it is not surprising that analytical solutions performed here can describe boundary layer flows.
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Figure 6: Trend of boundary layer velocity profile produced by analytical solution (solid line) and measured values (points) (Farrel and Iyengar 1999)
The next challenging case used for validation in this stage is the recently published combustion experiment due to Cuoci et al. (2007). The fuel is fed in a central tube (3.2 mm internal diameter and 1.6 mm wall thickness), centered in a 15 cm x 15 cm square test section, 1m long, with flat Pyrex windows on the four sides. The fuel molar composition is 39.7% CO, 29.9 H2, 29.7 N2 and 0.70 CH4. Ammonia was added in different amounts up to 1.64%; in the absence of ammonia, methane was not included in the fuel mixture. The average fuel flow velocity was 54.6 m/s with a resulting Reynolds number of ~8500; the inlet flow air velocity was 2.4 m/s. The inlet temperature of both streams is ~300K. Several radial profiles of velocity, temperature and species concentrations are available at different distances from the fuel inlet. As shown in Fig. 7, the analytical solution could reproduce the velocity change throughout the axial line with good agreement with the measured values. Detailed analysis for this case needs other equations (energy, species and thermodynamic state) to be solved simultaneously in order to describe turbulent-reaction interactions properly. This is of course a very challenging task and less tractable by considering that full mathematical theory for the Navier-Stokes equations is not yet complete. However, the comparison here is to show the potential of the simple analytical solution to tackle complex cases.
Figure 7: Measured mean axial velocity along flame centre line (points) (Cuoci et al. 2007) against the analytical solution (solid line) Int. J. of Appl. Math and Mech. 6 (11): 98-117, 2010.
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6 CONCLUSIONS It is concluded that the classes of solution (3a) and (7) drive the Vorticity and continuity equations to produce simple solutions. The original problem is transformed to the class of simpler parabolic and elliptic equations. An analysis of linear differential equations can be utilized to show that the solutions exist for L , L2 , Lp and H 1 which imply that H 3 to ensure the regularity. Moreover, the uniqueness problem is also analysed. Therefore, based on the condition discussed by Chae and Choe (1999), solutions (3a) and (7) will also satisfy Navier-Stokes equations. It is also important to mention here that the solution in (7) is weaker than (3a) since it needs further assumption in (9). Moreover, construction of classical solution in L , weak solution in L2 and strong solution in Lp for initial value problem for (12) can be developed easily like (10) since the bound of f is guaranteed. The basic analytical framework for turbulent free jets, boundary layer and combustions have also been presented. The governing equations based on the exact solutions to the incompressible Navier-Stokes equations are developed. It gives hindsight that turbulence closure can be achieved either by modeling or by exact solutions. Although there are different characteristic properties of flow, the predictions are shown to be in excellent agreement with experimental data. In fact, based on this physical insight, most of the characteristic properties could have been deduced by a priori reasoning alone within general equations of fluid dynamics. As the flow geometries become more complex, the analytical methods used in fluid dynamics research will also have to evolve. Engineers of computational fluid dynamics have much experience with complex geometries, and much can be learned about techniques from them. However, the significantly higher accuracy required by exact solutions must be kept in mind. Nonlinear methods of analysis and development are likely to prove to be very productive but still have to be developed more to reveal the secret of the Navier-Stokes equations.
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