2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems Timisoara, Romania October 24 - 26, 2007
Scientific Bulletin of the “Politehnica” University of Timisoara Transactions on Mechanics Tom 52(66), Fascicola 6, 2007
STABILITY OF THE STEADY FLOW IN ROTATIONALLY SYMMETRICAL DOMAIN František POCHYLÝ
Pavel RUDOLF *
V. Kaplan Department of Fluids Engineering, Brno University of Technology
V. Kaplan Department of Fluids Engineering, Brno University of Technology
Vladimír HABÁN
Jiří KOUTNÍK
V. Kaplan Department of Fluids Engineering, Brno University of Technology
Voith-Siemens Hydro Power Generation, Heidenheim
Klaus KRÜGER Voith-Siemens Hydro Power Generation, Heidenheim
*Corresponding author: Brno University of Technology, Technická 2, Brno, Czech Republic 61669 Tel.: +420541142336, Fax: +420541142347, E-mail:
[email protected] ABSTRACT Perturbation analysis with a linear approach of the flow in cylindrical domain was carried out in this paper. Analysis is based on Euler equations for inviscid fluid and continuity equation in rotating frame of reference. Eigen mode shapes of velocities visualize the flow patterns and eigen mode shapes of the cavitating vortex rope show origin of the elliptical vortex rope cross-section. Investigation confirms that boundary conditions have dominant effect on steady flow stability. Results indicate that careful design of the runner and the draft tube can lead to reduction of the instability consequences or to shifting the instability to different operational ranges. KEYWORDS Swirl, stability, rotationally symmetrical, eigen value, cylinder, draft tube, elliptical NOMENCLATURE ai bi h [Pa] hI hR L [m] p [Pa]
real part of the eigen value of velocity imaginary part of the eigen value of velocity eigen mode shape of pressure imaginary part of the eigen value of pressure real part of the eigen value of pressure length of the domain pressure
R s t u v wi
[m]
[s] [m.s-1] [m.s-1] [m.s-1] w0i [m.s-1] w03 [m.s-1]
w0φ [m.s-1] x i [m]
vortex rope radius complex eigen value time eigen mode shape of velocity velocity perturbation relative velocity components steady part of relative velocity components steady part of axial relative velocity component steady part of circumferential relative velocity component inertial coordinate system
y i [m]
rotating coordinate system α real part of the eigen value Δ [m] perturbation of vortex rope radius δ [m] eigen mode shape of vortex rope radius deformation φ [rad] angular coordinate σ [Pa] pressure perturbation σ p [Nm-1] surface tension
Ω [rad.s-1] angular speed of rotation
ω
imaginary part of the eigen value
INTRODUCTION Flow in many engineering applications is accompanied by a swirl, e.g. combustors, cyclones, hydraulic turbine draft tubes. An instability occurs if the swirl
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Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems
intensity reaches certain level. This instability is usually called vortex breakdown. First, vortex breakdown manifests as a symmetrical instability with axial flow reversal along axis and with further increase of the swirl transforms into helical form. This stage is well known in hydraulic machinery as a vortex rope. Its existence is connected with generation of pressure pulsations, which are a potential danger for turbine operation. Therefore it is of a vital importance to understand, why and when vortex rope in the turbine draft tube appears. There were many attempts to depict the vortex breakdown phenomenologically [1], [2], [3], [4] and analytically [5], [6]. CFD community tackles the problem numerically, e.g. [7], [8], [9], [10], [11]. Despite this effort it is obvious that although decades of research were devoted to this problem there are still open questions, ambiguities and most of all practical guidelines for vortex rope avoidance are missing. New insights, enabled by mixture of analytical, numerical and experimental approaches, are therefore valuable and welcome. Presented paper would like to set up methodology for analytical solution of cavitating vortex rope stability conditions in the hydraulic turbine draft tube. This approach is inevitably connected with simplifying assumptions, therefore the conclusions are supported by numerical (CFD) computations. Analytical investigation, on the other hand, enables to gain deeper insight into the relations among decisive parameters and judge their influence on the overall flow situation.
Figure 1. Flow domain
BASIC EQUATIONS Aim of this paper to establish stability conditions for steady flow in rotationally symmetrical domain. Double connected region filled with liquid is assumed. Domain V is surrounded by boundary Γ = Γ1UΓ2 . Incompressible liquid enters the domain through surface S1 and leaves through surface S 2 . Situation is depicted in Fig.1, where n denotes unit outward normal vector respective to liquid. Coordinate system (xi ) is considered to be inertial
and coordinate system ( y i ) is rotating with angular
speed ω3 = Ω around axis x3 , see Fig.2. The resulting equations will be formulated in the rotating coordinate system ( yi ) , because the aim is to find angular speed influence on stability of the flow. Viscous forces effect will be neglected for simplicity. Euler equations written using Einstein summation symbolics are the starting point of the analysis:
δwi 1 ∂p + ε i 3k ε k 3mω32Ym + 2ε i 3k ω3 w3 + = 0 (1) δt ρ ∂yi
Figure 2. Coordinate systems and continuity equation for incompressible fluid: ∂wi =0 (2) ∂yi
δwi in eq.(1) represents substantial derivative: δt δwi ∂wi ∂wi = wj + (3) δt ∂yi ∂t where wi are relative velocity components, p is pressure and ε ijk is Levi-Civit antisymmetrical tensor. Influence of the boundary conditions will be further studied.
Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems
STABILITY CONDITIONS Stability conditions will be investigated using the procedure introduced by Taylor for study of flow between two concentric cylinders. This approach is based on superposing perturbations to basic steady flow w0i = w0i ( y j ) , p0 = p0 (y j ) : wi = w0i + vi ( y j , t ) ;
p = p0 + σ (y j , t )
(4)
vi is a velocity perturbation and σ is pressure perturbation. Relations (4) are inserted to fundamental equations (1), (2), small non-linear terms and steady terms are neglected. ∂v i ∂w0i ∂v 1 ∂σ + = 0 (5) v j + i w0 j + 2ε i 3k Ωv k + ∂t ∂y j ∂y j ρ ∂y i
∂vi =0 ∂yi
s ∫ ui ui* dV + ∫ V
+ 2ε i 3k Ω ∫ u k ui* dV + V
ui = a i + ibi ; h = hR + ihI ; ai , bi , hR , hI ∈ Re
v i ( y j , t ) = ui ( y j , s ) e
;
α ∫ ui ui*dV + ∫ V
σ = h (y j , s ) e
(7)
after inserting to relations (7) it is obvious that: vi = ui eαt (cos ωt + i sin ωt )
σ = heαt (cos ωt + i sin ωt )
(8)
where ω represents angular speed liquid self excited oscillations and α is connected with stability. α can acquire positive or negative values. α < 0 system is stable. If α = 0 stability limit α > 0 system is unstable. We have to keep in mind that results are obtained without the effect of viscous forces, which have damping effect. Inserting eq.(7) to eqs.(5), (6) leads to following eigen value problem: su i +
∂w 0 i ∂u 1 ∂h = 0 (9) u j + i w0 j + 2ε i 3k Ωv k + ∂y j ∂y j ρ ∂y i
∂ui =0 ∂yi
(10)
Qualitative analysis of the stability conditions will be performed now. Equations (9) will be multiplied by ui* (complex conjugate to ui ), which will introduce boundary conditions influence to mathematical model (9), (10). Result of multiplication:
(11)
(12)
Relation for α is obtained after inserting eq.(12) to eq.(11).
(6)
st
∂h * u dV = 0. ∫ ρ V ∂yi i 1
Equation (11) determines s as a function of given eigen mode shape. It can be split into real and imaginary parts. Real part determines α , imaginary part determines ω . Let us split ui and h into real and imaginary parts, because α will be of interest as it determines stability.
Eigen value and eigen mode shape analysis will be employed to study stability conditions. The eigen value problem is formulated from following assumption: st
V
∂w0i ∂u u j ui* dV + ∫ i ui* w0 j dV + ∂y j ∂y j V
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V
∂w0i ai a j + bi b j dV + ∂y j
(
)
+
⎡ ∂ ⎤ 1 (ai ai )dV + ∂ (bi bi )dV ⎥ + w0 j ⎢ ∫ ∂y j 2V ⎢⎣ ∂y j ⎥⎦
+
∂h ⎞ 1 ⎛ ∂hR ⎜⎜ ai + I bi ⎟⎟dV = 0 ∫ ρ V ⎝ ∂xi ∂xi ⎠
(13)
It is clear from eq.(13) that Corilios forces do not influence stability. Equation (13) can be simplified using GaussOstrogradskij theorem, continuity equation and impermeability conditions on boundary Γ(ci ni = 0) . Taking into account following equation: 2
u =u i ui* = a i a i + bi bi
(14)
we may write: 2
α ∫ u dV + ∫ V
+
V
∂w0i (ai a j + bi b j )dV + ∂y j
1 1 a w0 j n j dS + ∫ ρ 2V 2
∫ (hR a i ni + hI bi ni )dS = 0
(15)
S
S = S1US 2UΓ
(16)
Expression (15) enables to assess α based on boundary conditions and steady velocity w0i . It is convenient to replace components ai , bi by components a r , a φ , br , bφ , which determine radial and circumferential components of velocity eigen mode shapes. Let us introduce: y1 = r cos φ , y 2 = r sin φ w01 = wr cos φ − wφ sin φ w02 = wr sin φ + wφ cos φ
(17)
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Equation (15) can be written in following form after inserting eq.(17): 2
⎫⎪ ⎧⎪ ∂ ⎛ w0φ ⎞ ⎜⎜ ⎟⎟(a r aφ + br b φ )⎬dV + ⎪⎭ V ⎪ ⎩ ∂r ⎝ r ⎠ ∂w0φ (a 3 aφ + b3bφ )dV + a 32 + b32 dV + ∫ ∂y 3 V
α ∫ u dV + ∫ ⎨r V
+∫ V
∂w03 ∂y 3
(
2
(18)
Γ
S
+
Case II
)
+ ∫ a w0 j n j dS + ∫ (h R a r + h I br ) +
Both cases also indicate that Ω is not explicitly expressed in stability condition. The effect is indirect, influences α , because u depends on magnitude of Ω .
∫ (hR a 3 h3 + hI b3 n3 )dS
Double connected region is considered, see Fig.3. Constant pressure boundary condition is prescribed on boundary Γ1 . This condition implies h = 0 and hR = 0 ∧ hI = 0 . Condition for α is the same as in eq.(20) and conclusions about stability are also the same.
S1US 2
Equation (18) allows to judge stability for different boundary conditions. (2.14) Case I A) Single connected cylindrical region Γ1 = 0 is considered. Following conditions for steady flow are assumed: w0ϕ = 0 ⇒ liquid rotates as a solid body ( w0ϕ is relative velocity in rotating frame of reference!) w03 = c0 = const Boundary conditions: y3 = 0 : a r = 0 ; aφ = 0 ; a3 = 0 br = 0 ; bφ = 0 ; b3 = 0 y3 = L : hr = 0 ; hI = 0 in position r = R2 = const : a r = 0 br = 0 ; Stability condition (18) has following form now
α=−
1
∫u
2
dV
∫u
2
w03 dS 2
(19)
S2
V
It means that instability occurs for w03 < 0 , i.e. for axial flow reversal. B) Same boundary conditions as in previous case are considered with exception of boundary S, where following relation is prescribed between axial velocity and pressure:
σ = kv3 and hence: hr = ka 3 ; hI = kb3 ⎤ ⎡ 2 α=− ⎢ ∫ u w03dS 2 − k ∫ a32 + b32 dS1 ⎥ (20) 2 ⎥⎦ S1 ∫ u dV ⎢⎣S2
(
Case III More general case is considered now. Steady flow in single connected domain is prescribed by following boundary conditions: S1 :
w0φ = 0 ;
S2 :
p0 = konst
Γ2 :
w0 r = 0
)
V
It is apparent that steady flow is stable for case k < 0 and w03 > 0 . If w03 < 0 and k < 0 then instability occurs for certain values of w03 . Hence, magnitude of k influences stability of the basic steady flow.
w03 = c0
Same boundary conditions for unsteady components as in Case Ia are valid. Equation (18) has form: 2
α ∫ u dV + ∫ V
After inserting to stability condition (18): 1
Figure 3. Double connected domain
V
(
)
∂w03 2 2 a3 + b32 dV + ∫ u w03dS2 = 0 (21) ∂y 3 S 2
Pictures of steady velocity and pressure fields for two magnitudes of angular velocity Ω are presented in Figs. 4-9. They were acquired using inviscid simulations in Fluent software. It is apparent from the pictures that pressure decreases along the axis due to rotation of the liquid, which causes axial velocity drop, because of boundary condition assymetry on boundaries S1 , S 2 , where by assymetry a combination of velocity boundary
Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems
Flow Direction Ω = 8 rad.s-1, cax=1m.s-1
97
terms will be amplified or reduced by eigen mode
(
2
)
shapes u and a32 + b32 , see eq.(21).
Figure 4. Tangential velocities in meridional cross-section
x1 Figure 5. Static pressure in meridional cross-section
x3
Figure 6. Velocity vectors in meridional cross-section
Figure 10 .Axial velocities in meridional cross-section ( x1 = 0 ≡ draft tube axis )
Flow Direction Ω = 14 rad.s-1, cax=1m.s-1
Figure 7 Tangential velocities in meridional cross-section
x1
x3 Figure 8. Static pressure in meridional cross-section
Figure 9. Velocity vectors in meridional cross-section condition on inlet and pressure boundary condition on outlet is undestood. Figures 4-6 (angular velocity Ω = 8rad .s −1 ) depict situation before instability occurs, whereas Figs. 7-9 (angular velocity Ω = 14rad .s −1 ) show stability limit of the steady flow. Origin of the axially symmetrical vortex breakdown is well represented in Fig.8 with region of higher pressure (marked by white cross) that is associated with flow deceleration. ∂w03 < 0 in the axis Derivative of axial velocity is ∂y 3 vicinity, which has negative impact on stability. On the ∂w03 > 0 close to boundary Γ2 . These other hand ∂y 3
Figure 11. Circumferential and radial velocities projected onto meridional plane ( x1 = 0 ≡ draft tube axis ) It is clear from presented eigen mode shapes that ∂w03 2 a3 + b32 is negligible on boundary Γ2 and thus ∂y3 ∂w 2 2 it may be assumed that ∫ 03 a3 + b3 dV < 0 , which ∂ y 3 V
(
)
(
)
favors instability of the system. If w03 > 0 (for low Ω ) then
∫u
2
w03dS > 0 is a stabilizing term. Liquid starts
S2
to flow in opposite direction ( w03 < 0 ) in the axis vicinity when angular velocity Ω is increased and system looses stability. Axial velocities in meridional cross-section are depicted for this case in Fig.12.
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Stability condition (18) acquires following form ∂δ = 0 in position y 3 = 0 . under assumption of δ = 0, ∂y3 Because of low magnitudes of surface tension and small changes of w0φ :
α=
⎧ ∂w03 2 ⎫ a 3 + b32 dV ⎬ + ⎨− ∫ u dV ⎩ V ∂y 3 ⎭
(
1
∫
2
)
V
+
∫
⎧⎪ ⎫⎪ 2 ⎨− ∫ u w03 dS 2 ⎬ + u dV ⎪⎩ S2 ⎪⎭
∫
⎧⎪ w σ 03 p ⎨ 2 u dV ⎪⎩ 2 ρR0
∫
⎧⎪ σ p ⎨ 2 u dV ⎪⎩ 2 ρR0
1 2
V
+
Figure 12. Axial flow reversal at the cylinder outlet (Ω =16 rad.s-1) Case IV Double connected region is considered similar to Case II. The only difference is rotation of boundary Γ1 with respect to inertial system with angular velocity Ω . Boundary Γ1 is impermeable, radial velocities of the liquid and the rope are equal, surface tension effect is expressed by Laplace equation. Boundary conditions on Γ1 : wr =
∂Δ ∂Δ w03 + ∂y3 ∂t
(22)
⎛ ∂2Δ 1 ∂2Δ ⎞ ⎟ + 2 2 2 ⎟ ⎝ ∂y 3 R0 ∂φ ⎠
(23)
R1 = R0 + Δ
(24)
σ = σ p ⎜⎜
1
V
−
1
⎡⎛ ∂δ (L ) ⎞ 2 ⎤ ⎫⎪ ∫ ⎢⎢⎜⎜⎝ ∂ϕ ⎟⎟⎠ ⎥⎥dϕ ⎬ − 0⎣ ⎦ ⎪⎭
2π
(30)
2 ⎫⎪ ⎛ ∂δ ⎞ ∂w03 ⎟ ⎜ dld φ ⎬ ∫0 ∫0 ⎜⎝ ∂ϕ ⎟⎠ ∂y 3 ⎭⎪
2π L
V
It is obvious from (30) that conclusions of previous paragraphs remain valid. But stability can be significantly influenced by vortex rope shape. Stability conditions are identical with previous case ∂δ = 0 . If σ r ≠ 0 (i.e. without surface tension), if ∂ϕ then the eigen mode shape, which is changing with changing angle ϕ can influence stability. The lowest eigen mode shapes are those with biscuit or almost elliptical shape, see [13], [14]. The first three eigen mode shape are depicted in Figs. 13-15.
where R0 is the initial radius of the inner cylinder (Γ1 ) , Δ is deformation of boundary Γ1 , σ p is surface tension. Eigen value analysis is performed:
Δ = δe st ,
δ = δ R + iδ I
δ R , δ I ∈ Re
(25)
Boundary conditions (22), (23) are transformed : ar =
∂δ R w03 + αδ R − ωδ I ∂y3
(26)
br =
∂δ I w03 + αδ I − ωδ R ∂y3
(27)
⎡ ∂ 2δ 1 ∂ 2δ R ⎤ hR = σ P ⎢ 2R + 2 ⎥ R0 ∂φ 2 ⎦ ⎣ ∂y 3
(28)
⎡ ∂ 2δ 1 ∂ 2δ I ⎤ hI = σ P ⎢ 2I + ⎥ R0 ∂φ 2 ⎦ ⎣ ∂y 3
(29)
Figure 13. Zeroth eigen mode shape of radial velocity component in radial plane
Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems
Figure 14. First eigen mode shape of radial velocity component in radial plane
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Strong dependency on boundary conditions results from the analysis. Boundary condition, which combines pressure and axial velocity w03 can play an important role, see eq.(20). Quantity k depends on runner design and its magnitude can significantly influence stability of the steady flow and origin of the cavitating vortex rope. Instability responding to the first and the second eigen mode shapes arises in case that cavitating vortex rope defined by boundary Γ1 appears in domain V, see Figs. 14, 15. Cavitating vortex rope whirls around rotation axis with the first eigen mode shape and rotates around its own axis with the second eigen mode shape. The resulting motion is a superposition of these two eigen mode shapes. Amplitude of whirling motion and deformation increases due to the instability. Unfortunately these phenomena can only be described by non-linear theory. The main goal of this study was to shed light on the cause of instability ACKNOWLEDGEMENTS
Voith-Siemens Hydro Power Generation, Heidenheim is gratefully acknowledged for support of this research. REFERENCES
Figure 15. Second eigen mode shape of radial velocity component in radial plane CONCLUSIONS
Presented investigation points to main cause of instability origin - asymmetry of boundary conditions on boundaries S1 and S 2 . Consequences of the asymmetry can be influenced by shape of the boundary Γ2 (draft tube wall), but especially by distribution of circumferential velocity w0φ on boundary S1 . Stability of the steady flow also depends on distance of boundaries S1 and S 2 . It is apparent from eq.(18) that ∂ ⎛ w0φ ⎜ ∂r ⎜⎝ r
⎞ ⎟⎟ < 0 should be avoided. ⎠
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[10] Stein, P., Sick, M., Doerfler, P., White, P., Braune, A., 2006, "Numerical Simulation of the Cavitating Draft Tube Vortex in a Francis Turbine", Proc. 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama [11] Muntean, S., Ruprecht, A.. Resiga, R., 2005, "A Numerical Investigation of the 3D Swirling Flow in a Pipe with Constant Diameter. Part 1: Inviscid Computation", Proc. of Workshop on Vortex Dominated Flows Achievements and Open Problems, Timisoara, pp. 77-86 [12] Muntean, S., Buntić, I., Ruprecht, A.. Resiga, R., 2005, "A Numerical Investigation of the 3D Swirling Flow in a Pipe with Constant Diameter. Part 2: Turbulent Computation", Proc. of Workshop on Vortex Domi-
nated Flows - Achievements and Open Problems, Timisoara, pp. 87-96 [13] Koutník, J.,Krüger, K., Pochylý, F., Rudolf, P., Habán, V., 2006, "On Cavitating Vortex Rope Form Stability During Francis Turbine Part Load Operation" Proc. IAHR Int. Meeting of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Barcelona [14] Rudolf, P., Habán, V., Pochylý, F., Koutník, J., Krüger, K., 2007, "Collapse of Cylindrical Cavitating Region and Conditions for Existence of Elliptical Form of Cavitating Vortex Rope", Proc. IAHR Int. Meeting of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara
