Abstract
We present a numerical kinetic scheme for an unsteady mixed pressurized and free surface model. This model has a source term depending on both the space variable and the unknown U of the system. Using the Finite Volume and Kinetic (FVK) framework, we propose an approximation of the source terms following the principle of interfacial upwind with a kinetic interpretation. Then, several numerical tests are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- θ(x):
-
Angle of the inclination of the main pipe axis z=Z(x) at position x
- Ω(x):
-
Cross-section area of the pipe orthogonal to the axis z=Z(x)
- S(x):
-
Area of Ω(x)
- R(x):
-
Radius of the cross-section Ω(x)
- σ(x,z):
-
Width of the cross-section Ω(x) at altitude z
- p(t,x,y,z):
-
Pressure
- ρ 0 :
-
Density of the water at atmospheric pressure p 0
- ρ(t,x,y,z):
-
Density of the water at the current pressure
- \(\overline{\rho}(t,x) = \frac{1}{S(x)}\int_{\Omega(x)} \rho(t,x,y,z)\,dy\,dz\) :
-
Mean value of ρ over Ω(x) (press. flows)
- c :
-
Sonic speed
- \(\mathcal {\mathbf {S}}(t,x)\) :
-
“Physical” wet area i.e. part of the cross-section area in contact with water (equal to S(x) if the flow is pressurized)
- \(A(t,x) = \frac{\overline{\rho}(t,x)}{\rho_{0}} \mathcal {\mathbf {S}}(t,x)\) :
-
Equivalent wet area
- u(t,x):
-
Velocity
- Q(t,x)=A(t,x)u(t,x):
-
Discharge
- E :
-
State indicator. E=0 if the flow is free surface, E=1 otherwise
- \(\mathcal{H}(\mathcal {\mathbf {S}})\) :
-
The Z-coordinate of the water level equal to \(\mathcal{H}(\mathcal {\mathbf {S}})=h(t,x)\) if the state is free surface, R(x) otherwise
- p(x,A,E):
-
Mean pressure over Ω
- K s >0:
-
Strickler coefficient depending on the material
- P m (A):
-
Wet perimeter of A (length of the part of the channel section in contact with the water)
- \(R_{h}(A) = \frac{A}{P_{m}(A)}\) :
-
Hydraulic radius
- Bold characters:
-
are used for vectors, except for \(\mathcal {\mathbf {S}}\)
References
Bourdarias, C., Ersoy, M., Gerbi, S.: A kinetic scheme for pressurized flows in non uniform closed water pipes. Monogr. Real Acad. Cienc. Zaragoza 31, 1–20 (2009)
Bourdarias, C., Ersoy, M., Gerbi, S.: A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. Int. J. Finite Vol. 6(2), 1–47 (2009)
Dal Maso, G., Lefloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74(6), 483–548 (1995)
Ersoy, M.: Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince. PhD Thesis, Université de Savoie, Chambéry, France (2010)
Greenberg, J.M., LeRoux, A.Y.: A well balanced scheme for the numerical processing of source terms in hyperbolic equation. SIAM J. Numer. Anal. 33(1), 1–16 (1996)
Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4), 201–231 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bourdarias, C., Ersoy, M. & Gerbi, S. A Kinetic Scheme for Transient Mixed Flows in Non Uniform Closed Pipes: A Global Manner to Upwind All the Source Terms. J Sci Comput 48, 89–104 (2011). https://doi.org/10.1007/s10915-010-9456-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-010-9456-0