ZERO GRAVITY OF A JET EMERGING FROM A SLIT

  J. Indones. Math. Soc. (MIHMI) Vol. 12, No. 1 (2006), pp. 89–98.

  

ZERO GRAVITY OF A JET

EMERGING FROM A SLIT

_Abstract.

L.H. Wiryanto

A jet formed as a fluid flows through a slit between two rectangle walls is

presented as a model for the case of zero gravity. The analytical solution is obtained by

using hodograph transformation for the slope of the free boundaries of the jet. This is

then used to determine the relation between the coordinates of the separation point of

the vertical wall and the asymptotic angle of the jet.

  1. INTRODUCTION A 2-D steady flow passing through a slit and producing a jet is considered. At the beginning the fluid occupies a domain bounded by two rectangular walls, and a slit is built by shifting one of the walls such that a jet is formed as the fluid flows through the slit. The problem here is to determine the profile of the jet which is a stream between two free boundaries.

  In solving of the problem we assume that the fluid flow is irrotational and the fluid is inviscid and incompressible. Moreover, we simplify the model by considering the influence of gravity relative to inertia is negligible. Physically, the jet asymp- totes downstream to a uniform stream, which is straight and declines downward at the same unique angle, to be determined. An exact solution can be obtained via a hodograph transformation in determining the slope of both free boundaries of the jet, since the speed of the fluid flow is constant along the free boundaries. The similar works can be seen in Dias & Tuck [1] for a flow over a vertical weir, and further generalized by Wiryanto [5] for various angles of the weir. Other references Received 4 August 2005, Revised 12 October 2005, Accepted 18 October 2005. _{2000 Mathematics Subject Classification Key words and Phrases} : 76B07.

  : Zero gravity, jet emerging from a slit, hodograph transformation.

  

L.H. Wiryanto

  90 for zero-gravity works can be found in Goh [2] for a jet emerging from a nozzle, and Tuck & Vanden-Broeck [3] for ploughing flows.

  In this paper we solve the problem for the fluid domain which is rectangle and involving two separation points, as the change of the boundary from solid to free surface. When we choose one of the separation point as the reference of coor- dinates, the coordinates of the other point should be an input parameter. However the domain transformation requires the artificial parameter rather than physical coordinates. Therefore, we determine the relation between both parameters, and the asymptotic angle of the jet.

  In discussing the problem, we first formulate the model in artificial variables. This produces a boundary value problem of Laplace equation presented in Section

  2. The analytical solution of the slope of the free boundaries of the jet is obtained by introducing an analytic complex function and applying Cauchy theorem. We discuss the solution in Section 3. This is then followed by discussing the result in Section 4. Some numerical calculations are used to reformulate the result in physical domain.

  2. MATHEMATICAL FORMULATION Let us consider the steady two-dimensional fluid flow such as illustrated in

  Figure 1a. Two walls, given in bold line, bound the fluid, and make a slit at the bottom corner by shifting one of the walls. Here we assume that the volume of the fluid is infinity by expressing no boundaries at the left and above. Therefore, the flux Q of the flow has less velocity for the fluid far from the slit.

  Without loss of generality we choose the bottom wall as a fixed boundary and also as the reference of the horizontal axis. The vertical axis is chosen rectangle to the horizontal one and passing the separation point A of the bottom wall. The vertical wall is shifted upward and either right or left so that the coordinates of the separation point B of this wall is (L, W ).

  From the assumption for the fluid and the flow, we can present the stream in a complex potential f = φ + iψ, which corresponds to the complex velocity df /dz = u − iv. We other define the fluid flow domain in a physical plane as z = x + iy. For convenience, we work in non-dimensional variables by taking Q as the unit flux and W as the unit length, and we choose φ = 0, ψ = 0 at the separation point A. Therefore, the f -plane is a strip with width 1, also the width of the slit. Now, our task is to solve the boundary value problem

  2

  f = 0 ∇ in the flow domain, subject to the dynamics condition expressed by Bernoulli’s equation

  2

  

2

  u + v = 0 (1) Zero gravity of a jet

  91 Figure 1: (a) shows a schematic diagram of the non-dimensional physical z-plane, whilst (b)and (c) show the complex potential f -plane and the artificial ζ-plane respectively. along the free boundaries for zero gravity, and kinematics condition

  ∂φ = 0 (2)

  ∂~n where ~n is a normal vector of the boundaries. In determining φ (or ψ), we introduce a hodograph variable Ω = τ −iθ having a relationship df

  Ω

  = e (3) dz and an artificial plane ζ = ξ + iη satisfying

  1 log ζ. (4) f = −

  π The relation (4) represents a mapping of the flow domain from the f -plane to ζ- plane. This artificial plane is a half lower-plane with points A and B mapped to ζ = 1 and ζ = ξ b

  (< 0) respectively, and the end of the jet (φ = ∞) mapped to

  92 ^{L.H. Wiryanto} ζ = 0. The schematic diagram of the flow is shown in Figure 1. The hodograph variable Ω is then expressed as function of ζ, so that the boundary value problem is now to determine Ω satisfying

  iτ

  b

  , ₋

  iθ

  √

  (1−ξ)(ξ−ξ b )

  for ξ

  b

  < ξ < 1, ₋

  √

  (1−ξ)(ξ b

⁻

ξ)

  (ξ−1)(ξ−ξ b )

  for ξ > 1 (8)

  On the other hand, the analytic function χ tends to zero as |ζ| → ∞. There- fore, Cauchy’s integral formula can be applied to χ on a path consisting of the real ζ-axis, a semi-circular at |ζ| = ∞ in the lower half plane, and a circle of vanishing radius about the point ζ ^∗ . This is then taken Im(ζ ^∗

  ) → 0 ⁻ giving χ(ξ ^∗ ) = i

  π ^{Z ∞ −∞} χ(ξ)

  ξ − ξ ^∗ dξ (9) where ξ ^∗ = ξ ^∗ + i0.

  The tangential θ of the streamline along the free boundaries can be deter- mined by substituting (8) into both sides of (9). For any ξ ^∗ in (ξ b , 1), the left hand side of (9) reduces to −iθ/ p(1 − ξ ^∗ )(ξ ^∗

  − ξ

  for ξ < ξ

  √

  ∇

  1/2

  2

  Ω = 0 subject to τ = 0 for ξ b < ξ < 1 (5)

  θ = −π/2 for ξ < ξ b

  , for ξ > 1.

  (6) Condition (5) is obtained by substituting (3) to (1), and condition (6) expresses the kinematics (2) along the vertical and horizontal walls.

  3. ANALYTICAL SOLUTION In this section, we solve analytically Ω(ξ) satisfying the boundary value prob- lem derived in the previous section, more specific the value of θ along the free boundaries ξ

  b

  < ξ < 1. To do so, we define an analytic complex function, namely χ(ζ) = Ω(ζ)(1 − ζ) ⁻

  (ζ + ξ

  π/2−iτ

  b

  ) ⁻

  1/2

  (7) The square root behavior represents the flow near the separation points ζ = 1 and ζ = ξ

  b

  having θ = O(ζ ⁻

  1/2

  ), see Wiryanto [4]. The function (7) is then expressed along the real ζ-axis by substituting (5) and (6), giving χ(ξ) = ^{       } _{      }

  b ). Meanwhile, the right hand side of Zero gravity of a jet

  93 (9) contains a complex integrand in which the imaginary part is equated to the other side, giving _{Z ∗ dξ} ξ b

  −1 ∗ ∗ θ(ξ ) = )(ξ ) (10) p(1 − ξ − ξ b _{∞ ∗}

  2 )

  (ξ − ξ b − ξ) p(1 − ξ)(ξ The integral can be determined, using antiderivative _{hq i Z} (a+b)(x−a)

  2 arctan

  (a−b)(x+a)

  dx = + constant

  √ √

  2

  2

  2

  2

  x a − a (x − b) − b _∗

  b )/2, b = ξ b b )/2. Therefore, the

  where a = (1 − ξ − (1 + ξ )/2, x = ξ − (1 + ξ expression (10) becomes _{∗ ∗ ∗ " #} )(ξ b ) π r a + b r (1 − ξ − ξ

  θ(ξ (11)

  ) = − − arctan

  2

  2

  a

  2 − b a − b The result (11) is firstly checked for the value of θ near the separation point B. _∗

  In the artificial variable, we take ξ → ξ b . This gives b → −a having consequences _{∗ ∗}

  )(ξ ) r (1 − ξ − ξ b → 1,

  2

  2

  a − b r a + b

  → 0. _∗ a − b Therefore, θ(ξ _{∗ ∗} ) → −π/2 which is the direction of the stream on the vertical wall. _q

  a+b

  Similarly, for ξ → 1 we obtain θ(ξ ) → 0 as → ∞ or the arctan function

  a−b

  tends to π/2. Finally, the asymptotic direction of the jet is determined by evaluating _∗ (11) at ξ = 0, giving

  π θ + arctan

jet = − p|ξ b |.

  2 This shows that θ only depends on the quantity l = L/W , or ξ in artificial

  jet b variable, representing the distance of the vertical wall to the vertical axis.

  The relation between θ jet and l can be obtained by determining z(ξ) for the free boundaries of the jet. From expressions (3) and (4), we have ₋

  Ω

  dz −e

  = . (12) dξ πξ , 1) the real and the imaginary parts of (12) become

  Since τ (ξ) = 0 for ξ ∈ (ξ b dx − cos θ

  = , (13) dξ πξ dy

  − sin θ = . (14)

  

L.H. Wiryanto

  94 The value of z(ξ) is then evaluated by integrating (13) and (14). However, a major difficulty occurs with integration, in which the factor πξ in the denominator makes it essentially impossible to integrate through ξ = 0. Instead, we evaluate z(ξ) for

  S(ǫ, 1), where ǫ is a relative small number representing the uniform ξ ∈ (ξ b , −ǫ) stream in |ζ| < ǫ. Therefore, y and x are evaluated from _{ ∗ R}

  ξ ∗

_{ dξ, for ξ < ξ}

1 sin θ _{∗  π ξ b ξ} 1 − b < −ǫ y(ξ ) = _{ R}

  (15)

  1 ∗ _ 1 sin θ _∗ dξ, for ǫ < ξ < 1, _{ R} π ξ ξ 1 ∗ _{ ∗ dξ, for ǫ < ξ < 1} 1 cos θ _{∗ } π ξ ξ

  x(ξ ) = _{R −} (16)

  

ǫ

1 cos θ

^{ } x(ǫ) + x dξ, for ξ < ξ ^∗

• jet ∗ b < −ǫ

  π ξ ξ

  Here, we use value x as the consequence of truncating domain around

  jet

  ξ = 0, and we jump from one free boundary to the other. This value is determined geometrically, as illustrated in Figure 2.

  Figure 2: Geometry of x jet From the separation point A, we can calculate x(ǫ) and y(ǫ); and from point

  B we obtain y(−ǫ). Then, we make a line passing through the point (x(ǫ), y(ǫ)) and perpendicular to θ

  jet . The intersection between the line and y = y(−ǫ) is the

  where point for ξ = −ǫ. Therefore, x(−ǫ) = x(ǫ) + x jet y(−ǫ) − y(ǫ) x = .

  jet

  tan(θ + π/2)

  jet

  Finally, the distance of the vertical wall to the vertical axis is l = x(ξ ).

  b

  The sign of l indicates that the wall is on the right side of the vertical axis (positive) or on the other side (negative).

  Zero gravity of a jet

  95

  4. NUMERICAL CALCULATIONS The analytical solution θ(ξ) has been derived in the previous section. This function is then used to determine the coordinates (x, y) of the free boundaries of the jet. We calculate some points (x, y) numerically, since the integrations (15) and (16) are not simple. While they are obtained, the jet is performed as plot of them with a unique l and θ for each value ξ .

  jet b

  4.1. Numerical Procedure The numerical procedure is begun by discretizing (ξ S(ǫ, 1), where the

  b , −ǫ) first interval is divided into M subinterval and the second one into N subinterval.

  Each subinterval is defined by taking the same length of subinterval in φ, to get

  

1

  1

  better accuracy. If we define φ log(ǫ), the set of

  b = − log(|ξ b |) and φ ǫ = − π π

  subinterval end-points is ₋

  π(φ b +jdφ)

  , for j = 0, 1, . . . , M −e

  ξ = − −

  j π(φ ǫ (j−M −1)dφ)

  e , for j = M + 1, M + 2, . . . , M + N where dφ = (φ )/M and N = φ /dφ. For each point ξ , we denote θ = θ(ξ )

  ǫ − φ b ǫ j j j

  evaluated from (11), with θ = 0. These values are then used = −π/2 and θ M +N to approximate the integration in (15) and (16) by trapezoidal rules, to obtain the coordinates of the free surface of the jet.

  In applying the procedure we evaluate the values of x and y, for the bottom boundary of the jet, using 1 cos θ cos θ

  j+1 j

  x + + j = x j+1 (ξ j+1 j ), (17) − ξ

  2π ξ ξ

  j+1 j

  sin θ 1 j sin θ j+1

• y = y + (ξ ),

  

j j+1 j+1 − ξ j

  2π ξ ξ

  j j+1

  = 0 and y = 0 represent for j = M +N −1, M +N −2, · · · , M +1, where x M +N M +N the coordinates of the separation point on the bottom wall, that are chosen as the center coordinates.

  We then continue to evaluate the coordinates of the other free surface. The value of y is computed similar to (17), but it is started from the separation point on the vertical wall, i.e.

  1 sin θ sin θ

  j j−1

  y = y + (ξ ),

  

j j−1 − j − ξ j−1

  2π ξ ξ

  j j−1

  = 1 as the distance of the vertical wall to the bottom for j = 1, 2, · · · , M, and y one. When y is obtained, the value of x at the end of the jet, which is on the

  M

  upper free surface, is determined by x M = x M +1 + x jet

  

L.H. Wiryanto

  96 where y

  M − y M +1 x = . jet

  tan(θ + π/2)

  jet

  For the other values of x j , j = M − 1, M − 2, · · · , 0, we evaluate using (17). The last value of x represents the distance of of the vertical wall to the vertical axis l.

  4.2. Results Most of our calculations uses M = 100 grids and ǫ = 0.01. These numbers are able to give four-figure accuracy for l. This accuracy is observed following the method in Wiryanto & Tuck [6] for some various M .

  Figure 3: Plot of a jet emerging from a slit with l = 0.4687 Figure 4: Plots ξ versus l in (a) and l versus θ in (b)

  b jet

  Zero gravity of a jet

  97 As the result, we show in Figure 1(a) a typical jet emerging from a slit for the vertical wall being on the left vertical axis, l = −0.3636 (equivalent with ξ b = −10), producing θ

  jet = −17.55 degree. When we increase l, the jet emerges with smaller

  θ . This is shown in Figure 3 for l = 0.4687 (equivalent with ξ

  jet b = −2) to give

  a comparison to the previous calculation. We obtain that the jet emerges with smaller asymptotic angle θ

  jet = −35.26 degree.

  For some various ξ b we calculate l and θ jet , and we show plot of them in Figure 4. The value of l = 0, the vertical wall is on the vertical axis, is obtained for ξ

b = −4.541 and producing θ jet = −25.19. Plot of the jet is performed in Figure 5.

  Figure 5: Plot of jet emerging from a slit with l = 0

  5. CONCLUSION We have solved the extreme case of a flow producing a jet emerging from a slit, in which the effect of gravity is assumed to be negligible. The analytical solution is obtained in term of the tangential of the curve boundary of the jet, and the profile of the jet is performed by plotting some points calculated from an integration involving that analytical solution. We found that the jet emerges downward asymptotically to a uniform stream with angle θ jet , depending on the position of the vertical wall to the vertical axis l.

  Acknowledgement. The research of this paper was supported by Department of Mathematics Institut Teknologi Bandung, contract no. 016/K0.1.7.5/PPD/2005, and this is gratefully acknowledged.

  

L.H. Wiryanto

  98 _REFERENCES

  

1. F. Dias and E.O. Tuck, “Weir flows and waterfalls”, J. Fluid Mech. 320 (1991),

525-539.

  

2. K.H.M. Goh, Numerical solution of quadratically nonlinear boundary value problems

using integral equation techniques, with applications to nozzle and wall flows , Ph.D. thesis, Department of Applied Mathematics, The University of Adelaide, 1986.

  

3. E.O. Tuck and J.-M. Vanden-Broeck, “Ploughing flows”, Euro. J. Applied Math.,

₉ (1998), 463-483.

  

4. L.H. Wiryanto, Some nonlinear shallow-water flows, Ph.D. thesis, Department of

Applied Mathematics, The University of Adelaide, 1998.

  

5. L.H. Wiryanto, “Zero gravity of free-surface flow over a weir”, Proc. ITB 31 (1999),

1-4.

  

6. L.H. Wiryanto and E.O. Tuck, “A back-turning jet formed by a uniform shallow

stream hitting a vertical wall”, Proc. Int. Conf. on Dif. Eqs.: ICDE’96, E. Soewono _{L.H. Wiryanto} and E. van Groesen (eds.), Kluwer Academic Press, 1997.

  : Department of Mathematics, Jalan Ganesha 10, Institut Teknologi Bandung, Bandung 40132, Indonesia. E-mail: leo@math.itb.ac.id.