Luận văn Hydraulic modeling of open channel flows over an arbitrary 3-D surface and its applications in amenity hydraulic engineering

p J zyx 2221 term)(pressure To estimate the pressure terms in Equations (3.33), (3.34), the momentum equation in ζ -direction of Equation (3.25) is considered. If the acceleration and shear stress components are negligible, the equation can be reduced as: ( ) ρζ ζζζζζ ηη ζ ηξ ζ ξη ζ ξξ p JJ GVVVUUVUU J zzx ∂ ∂++ −=Γ+Γ+Γ+Γ 0 222 00 1 (3.35) therefore, ( )[ ]ζζηηζηξζξηζξξζζζρζ GVVVUUVUU p zzx +Γ+Γ+Γ+Γ− ++ = ∂ ∂ 222 1 ζ is a straight axis consequently, 1222 =++ zzx ζζζ , ( ) ζζηηζηξζξηζξξρζ GVVVUUVUUp +Γ+Γ+Γ+Γ−=∂∂ (3.36) The expression for pressure distribution is obtained by integrating Equation (3.36) with respect to ζ . It follows that: { } 2 2 0 0 000 hGVVUVVUUUdp h ζζ ηη ζ ξη ζ ξη ζ ξξζρ −Γ+Γ+Γ+Γ=∫ (3.37) Equations (3.26, 3.33, 3.34) and (3.37) are the depth-averaged equations in a general form used in calculating water surface profile of flows in the following applications. 29 Notations t : time zyx ,, : Cartesian coordinates ζηξ ,, : generalized curvilinear coordinates K,,,,,, zyxzyx ηηηξξξ : components of transformation matrix J : transformation Jacobian ( )wvu ,, : velocity components in Cartesian coordinate ( )WVU ,, : contravariant components of velocity vector ( )sss WVU ,, : contravariant components of velocity vector at water surface h : water depth in ζ -direction p : pressure ρ : water density g : gravitational acceleration ζηξ GGG ,, : contravariant components of gravitational vector Kyyyxxzxyxx τττττ ,,,, : effective shear stress ηξ ττ bb , : contravariant of shear stress acting on the bottom i jkΓ : Christoffel symbol 30 Chapter 4 STEADY ANALYSIS OF WATER SURFACE PROFILE OF FLOWS WITH AIR-CORE VORTEX AT VERTICAL INTAKE 4.1 Introduction Air-core vortex formation at intakes is a significant hydraulic engineering problem in many situations such as intakes for irrigation, drainage system, hydropower generation in which the water is normally drawn from rivers, channels or reservoir,… as well as in water art-works or sculptures. It occurs typically whenever the submergence is less than a critical value and causes some detrimental effects as reduction in intake discharge, resulting vibrations and noises as well as operational difficulties. Figure 4.1 shows an example of free surface air-core vortex occurred in laboratory. When the flow in a large body of slowly moving water is diverted and locally accelerated or drawn off, any associated vortex tube is extended and its rotation is thereby increased. Higher velocities incur lower pressures and, if a free surface exists, it becomes locally 31 Figure 4.1 An example of free surface air-vortex Figure 4.2 Various stages of development of air-entraining vortex: S1>S2>S3>S4 (Jain et al., 1978) Q Q S1 S2 S4 Q S3 a. No depression on the surface at large submergence b. Formation of a dimple d. Air-entraining vortexc. Air core extends deeper as the vortex becomes stronger 32 depressed. Thus in hydraulic structures, where flow negotiates a vertical shaft, submerged conduit or gated outlet, the vortex may be sufficiently intense to form a hollow core, which could transmit air. This may lead to the undesirable effects already mentioned, and considerable efforts have been made to predict and control such phenomena. Indeed, the deliberated introduction of a vortex at the vertical intake has been described in Jain et al. (1978). Figure 4.2 shows the various stages in the development of an air-entraining vortex when the water depth is decreased gradually. At the first stage, when the submergence is large, there is no depression on the surface (Figure 4.2a), but together with decreasing water depth, a dimple forms as in Figure 4.2b. If the submergence is further decreased the air-core vortex occurs (Figure 4.2c) and Figure 4.2d shows critical condition under which a vortex just tends to entrain air (Jain et al. 1978). Suppose that steady flow occurs in the horizontal yx − plane and that closed path, S , is traced out in the fluid (Figure 4.3). The path encloses an area A by linking adjacent flow particles and subsequently moves with the flow. At a point on the path, one particle may have velocity components nV and tV normal and tangential to the path. For the path to remain unbroken and assuming the flow is incompressible, the sum ∑ ∆SVn is the net inflow into the closed area A and must always be zero. The corresponding sum along the path ∑ ∆SVt is not necessarily zero and is known as the circulation Γ (being defined as positive anticlockwise). The Kenvin’s theorem states that the circulation remains constant with time unless an external shear stress exists along S , as described in Townson (1991). 33 Figure 4.3 The inflow to and circulation round a closed path in a flow field (Townson 1991) Figure 4.4 The concept of simple Rankine vortex including two parts: free vortex in outer zone and forced vortex in inner zone (Townson 1991) S A nV S∆ tV r ∞→r 0→tV H h H∆ free vortex forced vortex ideal fluid real fluid 34 If we consider an infinitely wide reservoir of constant depth is drawn off at a central point with rate q per unit depth. At some distance from the center, a circular vortex tube may be defined with circulation Γ about a vertical axis through the drawn-off point. As the tube contracts to radius r under the influence of drawn-off, the tangential velocity rVt π2/Γ= . The radial velocity is rqVr π2/= , and both clearly accelerate, producing a spiral flow towards the center. Applying the Kelvin’s theorem, the circulation is conserved therefore the velocity at the center is infinite. If the pressures remain hydrostatic, the depression of the free surface is also infinite. But the viscosity present in real fluids prevents this condition arising, and a zone in center of the flow rotates as a solid mass (Townson 1991). This central part is known as a forced vortex and the composite system as Rankine’s vortex (Figure 4.4). Several approaches have been presented in the literature to deal with the problem of determination and prediction of critical submergence serving in design works. These approaches basically can be labeled as analytical models (Jain 1984; Odgaard 1986; Hite and Mih 1994) and physical models (Anwar et al. 1978; Jain et al. 1978; Yildirim and Kocabas 1995; Yildirim and Kocabas 1997; Yildirim and Kocabas 1998). Many analytical attempts have been presented in the literature in order to attain a theoretical view of the far-field velocity; in fact the flow representation has not been defined so far by any comprehensive analytical analysis. The concept of simple Rankine vortex normally used in the basic equations (Odgaard 1986; Hite and Mih 1994), therefore this approach obviously could not be applied for the case of air-entraining vortex and moreover, the Kelvin’s theorem is invalid for the central region. Trivellato et al. (1999) set the water surface equal to the stationary headwater while other experimental works only focused on the critical submergence. Consequently, these approaches could not be used to predict the water surface profile of flow with air-core 35 vortex. In this study, the water surface profile of a steady air core vortex flow into a vertical intake was derived through out a depth-averaged model of open channel flows over the 3-D curvilinear bottom using a generalized and body fitted coordinate system. The assumption of fully free air-core vortex in the new coordinate allowed us to use the Kelvin’s theorem of the conservation of circulation for the whole flow field. The vortex was assumed axisymmetric and steady. The assumption of shallow water and kinetic boundary condition at water surface were also used. The equation describing water surface profile was derived and calculated results were compared to the formula introduced by Orgaard (1986). The application’s results showed us the ability of the model in analyzing the water surface profile and can be improved to simulate the flow structure of an air-core vortex. 4.2 Governing equations Coordinate setting To consider the vortex occurring at a cavity on the bottom surface (Figure 4.5), the position of any point say P , is defined by three coordinates ( )ζηξ ,, where ( )ηξ , define the position of 0P (projection of P ) on the bottom, and ζ is the distance from point P to that bottom surface. Assuming that the shape of bottom surface (i.e at 0=ζ ) has the form of )( 0 0 ar bz − − = , (4.1) where a and b are the coefficients which define the shape of the bottom; 0r is the distance from any point on the bottom to the z -axis 2 0 2 00 yxr += (4.2) 36 Figure 4.5 Definition of coordinate components x r y η a O ξ ζ P P0 r x ad 2= aa− • z l • 37 Based on Figure 4.5, we can derive the relations ηcos00 rx = and ηsin00 ry −= . Then, the bottom surface is expressed by the following equation: 0)(),,( 00000 =+−=Φ bzarzyx (4.3) The unit vector )(n normal to the bottom surface is derived as: k arz ar j r zy arz i r zx arzgrad gradn rrrr 2 0 2 0 0 0 00 2 0 2 00 00 2 0 2 0 )()( 1 )( 1 −+ − + −+ + −+ = Φ Φ = At ζ the relations between ),,( zyx and ),,( ζηξ can be expressed as: ζηηζ 24 0 0 0 00 2 0 2 0 0 )( coscos )( 1 bar br r zx arz xx +− −= −+ += (4.4a) ζηηζ 24 0 0 0 00 2 0 2 0 0 )( sinsin )( 1 bar br r zy arz yy +− +−= −+ += (4.4b) ζζ 24 0 2 0 020 2 0 0 0 )( )( )()( bar ar ar b arz ar zz +− − + − −= −+ − += (4.4c) Then the covariant base vector components on the bottom surface are denoted as follows: k arp bj p i p kzjyixe 2 0111 000 )( sincos − −+−=++= ηη ξξξξ rv (4.5a) jrirkzjyixe ηηηηη ηη cossin 00000 −−=++= rr (4.5b) j bar bi bar bkzjyixe rrr 24 0 24 0 000 )( sin )( cos +− + +− −=++= ηη ζζζζ k bar ar r 24 0 2 0 )( )( +− − + (4.5c) Let’s denote that 2 0 2 0 22 0 2 0 00 2 0 1 )(2)()( )()( barlbarlarr brarbarl p +−+−+− +−+− = (4.6) 38 in which l is defined in Figure 4.5. From Figure 4.5, it could be easily seen that ( ) ( )arr barl r ar bl r zl − +− = − + = − = 00 0 0 0 0 0tanξ (4.7) then ( ) ( ) 2020 00 2 0 00 0 00 )( )()(tan arr brarbarl arr barl rr − +−+− −=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − +− ∂ ∂ = ∂ ∂ ξ (4.8) From (4.6) we can obtain: ( ) ( ) ( ) ( )2020 2 0 2 0 22 0 2 02 2 2 tan1 cos 1tan arr barlbarlarr − +−+−+− =+== ∂ ∂ ξξξ ξ (4.9) but 0 00 0 tan tan tantan r rr r ∂ ∂ ∂ ∂ = ∂ ∂⇒ ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ξ ξ ξ ξξ ξ ξ ξ (4.10) Thus, substituting (4.8) and (4.9) into (4.10) gives 100 2 0 2 0 2 0 22 0 2 00 1 )()( )(2)()( pbrarbarl barlbarlarrr −= +−+− +−+−+− −= ∂ ∂ ξ (4.11) From (4.4a): ( ) ( )[ ] ζ ηη 2324 0 3 0 0 cos2 cos bar arb r x +− − += ∂ ∂ (4.12) Using the relations in (4.11), (4.12), it can be obtained: ( ) ( )[ ] ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −⋅⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ +− − += ∂ ∂ = 1 2324 0 3 0 1cos2cos pbar arbxx ζηηξξ ( ) ( )[ ] ζ ηη ⋅ +− − −−= 2324 01 3 0 1 cos2cos barp arb p (4.13) 39 Similarly, ( ) ( )[ ] ζ ηη ξ ⋅ +− − += 2324 01 3 0 1 sin2sin barp arb p y (4.14) ( ) ( ) ( )[ ] ζξ ⋅+− − − − −= 2324 01 0 2 2 01 2 barp arb arp bz (4.15) and ( ) ζ ηηη ⋅ +− +−= 24 0 0 sinsin bar brx (4.16) ( ) ζ ηηη ⋅ +− +−= 24 0 0 coscos bar bry (4.17) 0=ηz (4.18) ( ) 240 cos bar bx +− −= η ζ (4.19) ( ) 240 sin bar by +− = η ζ (4.20) ( ) ( ) 240 2 0 bar ar z +− − =ζ (4.21) and the Jacobian right on the bottom surface is: 2 0 24 00 100 )( )(1 11 000 000 000 ar barr p zzz yyy xxx JJ − +− === = ζηξ ζηξ ζηξ ζ (4.22) The gravitational vector can be expressed in the new coordinate system as follows: ζ ζ η η ξ ξ eGeGeGG ++= (4.23) Substituting the base vector components from (4.5) into (4.23) 40 ( ) ( ) ( ) kzjyixGkzjyixGkzjyixGG ζζζζηηηηξξξξ ++++++++= ( ) ( ) ( ) kzGzGzGjyGyGyGixGxGxG ζζηηξξζζηηξξζζηηξξ ++++++++= but kgG −= consequently, gG G G zzz yyy xxx − =⋅ 0 0 ξ η ξ ζηξ ζηξ ζηξ (4.24) hence, the contravariant components of gravitational vector are derived ( ) ( )( ) 240 2 0 1 bar ar bgpyxyxgJG +− − =−−= ηζζη ξ (4.25) ( ) 0=−−= ξζζξη yxyxgJG (4.26) ( ) ( )( ) 240 2 0 bar ar gyxyxgJG +− − −=−−= ξηηξ ξ (4.27) Using Equations (4.13-4.22) in plugging with (3.15-3.17), it gives: ( ) ( ) 240 4 0 1 cos bar ar px +− − −= ηξ (4.28) ( ) ( ) 240 4 0 1 sin bar ar py +− − = ηξ (4.29) ( ) ( ) 240 2 0 bar arb z +− − −=ξ (4.30) 0 sin rx ηη −= (4.31) 0 cos ry ηη −= (4.32) 41 0=zη (4.33) ( ) 240 cos bar b x +− −= ηζ (4.34) ( ) 240 sin bar b x +− = ηζ (4.35) ( ) ( ) 240 2 0 bar ar z +− − =ζ (4.36) Equation of water surface profile Dealing with the flows at a vertical intake as in Figure 4.5, the depth-averaged equations (3.26, 3.33 and 3.34) are applied. With the assumptions of steady vortex ( 0= ∂ ∂ t ) and axisymmetric flow, i.e. the flow is homogenous in η -direction ( 0= ∂ ∂ η ), those equations become: Continuity equation: 0 0 = J M d d ξ or .00 constQJ M == (4.37) Momentum equations: ξ -direction: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UM d d b ρ τ ξ ξξξ ηη ξ ηξ ξ ξη ξ ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ({ ) ⎭⎬ ⎫ −Γ+Γ+Γ+Γ ++ − 2 2 0000 0 2 0 2 0 2 0 hGVVUVVUUU d d J zyx ζζ ηη ζ ξη ζ ξη ζ ξξξ ξξξ (4.38) η -direction: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UN d d b ρ τ ξ η ηη ηη η ηξ η ξη η ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ 42 ({ ) ⎭⎬ ⎫ −Γ+Γ+Γ+Γ ++ − 2 2 0000 0 000000 hGVVUVVUUU d d J zzyyxx ζζ ηη ζ ξη ζ ξη ζ ξξξ ηξηξηξ (4.39) Using the definition in Equation (3.32) in plugging with Equations (4.28 – 4.36), the Christoffel symbols are hereafter calculated: ( ) ( )[ ]24001 2 0 1 2 1 0 21 bararp b dr dp p zyx zyx +−− += ∂ ∂ − ∂ ∂ − ∂ ∂ −=Γ ξ ξ ξ ξ ξ ξ ξξξ ξ ξξ (4.40) 000 =Γ=Γ ξ ηξ ξ ξη (4.41) ( ) ( ) 240 4 00 10 bar arr p +− − =Γ ξηη (4.42) 000 =Γ=Γ η ηη η ξξ (4.43) 01 00 1 rp −=Γ=Γ ηηξ η ξη (4.44) ( ) ( ) 2400210 2 bararp b +−− −=Γ ζξξ (4.45) 000 =Γ=Γ ζ ηξ ζ ξη (4.46) ( ) 240 0 0 bar br +− −=Γ ζηη (4.47) ( ) ( ) 240 4 0 2 12 0 2 0 2 0 bar arp zyx +− − =++ ξξξ (4.48) 2 0 2 0 2 0 2 0 1 rzyx =++ ηηη (4.49) ξ and η introduced in Figure (4.5) are obviously perpendicular with each other, thus it is given that 0000000 =++ zzyyxx ξηξηξη 43 Therefore, Equation (4.39) becomes: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UN d d b ρ τ ξ η ηη ηη η ηξ η ξη η ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ (4.50) By substituting Equations (4.26, 4.43-4.44) and neglecting the effect of friction in η -direction, Equation (4.50) is reduced as: 0 2 0100 =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ rphJ MN J MV d d ξ (4.51) Applying Equation (4.37) into (4.51) gives ( ) 02 01 00 =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+ rph NQV d dQ ξ or 0 0 01 2 2 r dr V dV rp V d dV −=⇒=ξ (4.52) Integrating Equation (4.52) between two arbitrary points A and B with distance Ar0 and 0Br respectively from z -axis, the following relation is obtained: 2 0 2 0 0 02 B A A B B A B A r r V V r dr V dV =⇒−= ∫∫ (4.53) where AV and BV are the contravariant components of velocity vector at A and B respectively. Based on equation (4.53), the depth averaged tangential velocity component at 0r is defined as: BBAA B A A A B B rvrv r r r v r v VreVv 002 0 2 0 0 0 0 =⇒=⇒== ηr (4.54) Eq. (4.54) is identical to the free vortex condition which allows us to assume the conservation of circulation vr02π=Γ for the whole flow regime, thus Vr 2 02π=Γ and 44 from which: 2 02 r V π Γ = (4.55) The evaluation of the contravariant component of shear stress acting on the bottom is approximated using the following relation: 2 0 2 0 0 2 hr Qf J UefUf bb =⇒== ρ τ ρ τ ξ ξ ξ rrV (4.56) Using the relations described in Equations (4.40-4.49) and (4.45-4.46), Equation (4.38) becomes ρ τ ξ ξ ξξ ξξ 00 0 0 2 0 2 J G J h hJ M hJ M b −=Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ( ) ⎥⎦ ⎤⎢⎣ ⎡ −Γ+Γ ∂ ∂ ++− 222 1 2 0 2 0 2 2 0 2 0 2 0 0 hGNM J zyx ζζ ηη ζ ξξξξξξ (4.57) Substituting (4.37) into (4.57) gives: ρ τ ξ ξ ξξ ξξ 00 0 0 2 0 0 2 0 1 J G J h h JQ hJ Q b−=Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ( ) [ ]ζξξξξξξ 02020200 2 0 2 Γ ∂ ∂ ++− zyxJ Q ( ) ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ Γ ∂ ∂ ++ Γ − ζ ηηξξξξπ 040 2 0 2 0 2 0 0 2 2 8 rJ zyx ( ) [ ]2202020 02 1 hG J zyx ζ ξξξξ ∂ ∂ +++ Then, using the relations (4.56) and after some manipulation, we obtain the equation for water surface profile in the form of ),( ),( 021 01 rhfp rhf d dh =ξ (4.58) in which 45 -0.1 0.0 0.1 0.2 x (m) -0.2 0.0 0.2 z (m ) Water surface Critical depth line Quasi-normal depth line Bottom Figure 4.6 An example of computed water surface profile withç quasi-normal depth line and critical depth line 46 [ ] [ ] [ ] 4 2324 0 2 0 3 00 24 0 2 2 2324 0 0 2 02 01 )( )(2)(3 8 )( )( ),( h barr arrbar b bar arr gbrhf ⎪⎭ ⎪⎬ ⎫ +− −++−Γ − ⎪⎩ ⎪⎨ ⎧ +− − = π [ ] [ ] [ ] 2 2524 0 24 0 2 02 02324 00 3 02 0 )( )()(3 )( )(2 h bar barar bQ barr ar bQ ⎪⎭ ⎪⎬ ⎫ +− −−− + ⎪⎩ ⎪⎨ ⎧ +− − + [ ] 2 0 2124 02 0 0 2 03 0 2 2 2 0 2 0 )( )( .- 1 4)( gb ar bar Qfh r Q h rar r − +− −⎪⎭ ⎪⎬⎫⎪⎩ ⎪⎨⎧ Γ− − + π (4.59) and [ ] [ ] 2 0 3 2124 00 2 2 3 2124 0 2 0 2 0 02 )( 1 4)( )( ),( Qbh barr gh bar arr rhf − +− Γ + +− − = π (4.60) Method of Calculation For convenience of simulating the water surface profile, the transformation of special coordinates from ξ to 0r is needed. Substitute Equation (4.11) into (4.58) it follows that ),( ),( ),( ),(1. 02 01 0021 01 01 0 0 rhf rhf dr dh rhfp rhf dr dh pd dr dr dh d dh −=⇒=−== ξξ (4.61) A common method of analysis including singular point often used in hydraulic engineering is applied. The singular point is the point at which both functions ),( 01 rhf and ),( 02 rhf in Equation (4.58) are equal to zero. The equation 0),( 01 =rhf expresses the quasi-normal depth line whereas 0),( 02 =rhf expresses critical depth line as shown in Figure 4.6. The water surface profile is derived from Equation (4.58) by using the fourth-order Runge-Kutta scheme with the initial slope near the singular point defined by the following equation: 47 S SSSSSS S h f r f h f h f r f h f r f dr dh ∂ ∂ ∂ ∂ ∂ ∂ −⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ ±⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ − = 2 0 12 2 1 0 21 0 2 0 2 4 (4.62) (The subscript “s” denotes the derivatives at singular point) 4.3 Results and discussions Posey and Hsu (1950) and Jain et al. (1978) have reported that a large reduction in intake discharge was due to formation of vortices, especially in case of flow entering the intake with entrapped air. The model herein derived has been applied with different imposed circulations (i.e. different strengths of vortex) and the computed results show reduction of discharge when the circulation increases while keeping the same submergence (water head) (Figure 4.7). It is observed that the larger the circulation the larger the air core of the vortex extends into the intake, and in case of air coming into intake the intake discharge decrease significantly (from sm331031.0 −× , in case TC01 to sm331006.0 −× , in case TC04 as in Table 4.1). This phenomenon is in agreement with the previous experiments (Jain et al. 1978; Posey and Hsu 1950). Intake discharge is inversely proportional to circulation as shown in Figure 4.8 where the circle marks are the calculated results and solid line is the trend line calculated by the least squares method. The results also show that for small decrease in submergence, the air core becomes larger (Figure 4.9). Similar results were reported by Anwar et al. (1978). In Figure 4.9, it is noted that when the circulation increases, the submergence also would increase to maintain the same intake discharge (see Table 4.2 for reference). Based on these results, it is evident that the model is capable of representing the effects of circulation on the flow through vertical intake. 48 Table 4.1 Parameters used in the calculations of results in figure 4.7 Case a (m) b (m2) Q0ç (10-3m3/s) Γ (m2/s) f TC 01 0.025 10-4 0.310 0.182 0.015 TC 02 0.025 10-4 0.230 0.200 0.015 TC 03 0.025 10-4 0.135 0.225 0.015 TC 04 0.025 10-4 0.060 0.250 0.015 -0.05 0.00 0.05 0.10 x (m) -0.05 0.00 0.05 0.10 0.15 z (m ) +++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + TC 01 TC 02 TC 03 TC 04 Bottom Figure 4.7 The effect of circulation on water surface profile andç discharge at the intake with same water head 49 Table 4.2 Parameters used in the calculations of results in figure 4.9 Case a (m) b (m2) Q0ç (10-3m3/s) Γ (m2/s) f TC 12 0.02 10-4 0.15 0.30 0.015 TC 13 0.02 10-4 0.15 0.25 0.015 TC 14 0.02 10-4 0.15 0.20 0.015 0.3 0.4 0.5 0.6 0 0.001 0.002 0.003 0.004 Γ (m2/s) Q (m 3/ s) Figure 4.8 Variation of intake discharge with circulation (a=0.025m, b=10-5 m2, water head=0.5m) -0.1 0.0 0.1 0.2 x (m) -0.1 0.0 0.1 0.2 z (m ) TC 12 TC 13 TC 14 Bottom Figure 4.9 Different water surface profiles with different values of circulationç while maintaining the constant intake discharge 50 The effects of the shape of intake have been examined by changing the value of ''b in equation (4.1). In this case, three simulations were considered with values of 54 10 5 ;10 −− ×=b and 2510 m− while maintaining the radius of intake (i.e value of ''a in Equation 4.1) at m2102 −× . It can be seen from Figure 4.10 that the water depth increases when the intake entrance becomes sharper (decreasing value of b as shown in Table 4.3). For more clarity, the computation has been done with the same intake and circulation to test the effects of value of b on intake discharge without changing the submergence (Figure 4.11a) and the effects on submergence with same discharge (Figure 4.11b). This phenomenon is consistent with the physical meaning of the coefficient ''b and it also improves the performance of bell-mouth intake. So far we have been able to verify the model qualitatively based on the fact that no previous studies are available which describe the variation in water surface profile at the intake. However, a larger quantity of experimental data has been used by Odgaard (1986) to verify the equation representing the critical submergence, which is defined as the submergence when the tip of air-core vortex just reaches the intake (Figure 4.12), in the absence of surface tension (Eq. 18 in Odgaard, 1986) presented as follows: [ ] 5.05.0)(6.5 2.23 FrN R d S e Γ= (4.63) where circulation function 0)( QSN Γ=Γ , Froude number ( ) 21gdVFr = and Reynold number dQ υ=Re . This equation can be applied for the case of vortex with turbulent core, in which υ is replaced by Γ+ kυ . From calibration the value of k was then estimated to be 5106 −×≈k . 51 Table 4.3 Parameters used in the calculations of results in figure 4.10 Case a (m) b (m2) Q0ç (10-3m3/s) Γ (m2/s) f TC 15 0.025 1.x10-4 0.3 0.225 0.015 TC 16 0.025 5.x10-5 0.3 0.225 0.015 TC 17 0.025 1.x10-5 0.3 0.225 0.015 0.00000 0.00006 0.00012 b (m2) 0.0000 0.0005 0.0010 0.0015 0.0020 Q (m 3/ s) 0.00000 0.00004 0.00008 0.00012 b (m2) 0 5 10 15 20 25 H /d Figure 4.11 The effects of b on discharge (17a) and submergence (17b) at an intake -0.1 0.0 0.1 0.2 -0.10 0.00 0.10 0.20 z (m ) TC 15 TC 16 TC 17 Bottom Figure 4.10 Changing of water surface profile with different shape of the intake 52 x z S d Figure 4.12 Definition sketch of critical submergence 100 101 102 Computed Critical Submergence (S/d) 100 101 102 O dg aa rd 's C rit ic al S ub m er ge nc e (S /d ) Figure 4.13 Comparison of computed critical submergence by the model (Equat