FLUID flow in a slightly inclined rectangular open

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1 Numerical Solution of de St. Venant Equations with Controlled Global Boundaries Between Unsteady Subcritical States Aldrin P. Mendoza, Adrian Roy L. Valdez, and Carlene P. Arceo Abstract This paper aims to verify numerical existence of boundary controls that steer the de St. Venant system in finite time, from a iven unsteady subcritical state to another. The method of characteristics is used in obtainin the numerical solution. The problem is divided into two parts: first, an unsteady subcritical flow is steered towards a steady one then such flow is steered towards another unsteady subcritical state. Index Terms lobal controllability first order quasilinear hyperbolic partial differential equation de St. Venant equations. I. INTRODUCTION FLUID flow in a slihtly inclined rectanular open channel with friction is modeled by the de St. Venant equations. If the open channel is defined lenthwise by x [0, L], the averae velocity of flow by vx, t, depth of flow by hx, t, θ as the slope of the open channel, s f for the coefficient of friction, and for the ravitational acceleration, then the de St. Venant equations can be written in the followin form: { v x h h x v h 0, 1 x h v x v v θ s f Equations in 1 are also known as continuity and momentum equations, respectively. The continuity equation is established by equatin the net inflow and the rate of chane of fluid in the control volume. On the other hand, the momentum equation is derived by applyin the second law of motion on the fluid inside the control volume. The control volume is a fixed volume in space wherein fluid may flow in and out. It is taken as volume from infinitesimal lenth of the channel dx over a small period of time. The de St. Venant equations can be written in a variety of form. It can be written in matrix form as h v v h v x h v 0 θ s f. 2 Manuscript received December 08, 2010 revised January 06, This work is supported in part by the Commission on Hiher Education CHED-Philippines under Hiher Education Development Project-Faculty Development Proram HEDP-FDP Scholarship Grant. A.P. Mendoza is a raduate student of the Institute of Mathematics, University of the Philippines, Diliman, Quezon City, 1101, on study leave from the Institute of Enineerin, Tarlac Collee of Ariculture, Camilin, Tarlac, 206 Philippines aldrin.mendoza@up.edu.ph. A.R.L. Valdez is with the Scientific Computin Laboratory, Department of Computer Science, University of the Philippines, Diliman, Quezon City, 1101 Philippines alvaldez@dcs.upd.edu.ph C.P. Arceo is with the Institute of Mathematics, University of the Philippines, Diliman, Quezon City, 1101 Philippines cayen@math.upd.edu.ph The partial differential equations can be reduced into ordinary differential equations known as the characteristic form of the de St. Venant equations, written as follows: dv ± 2c v ± 2c dx v ± 2c θ s f, x t v ± c, where ct, x hx, t 1 2 is the wave celerity. They can also be written as a first order quasilinear hyperbolic system of diaonal form R R A R, R x R R θ sf, 4 θ s f where R v2c, R v 2c are the Riemann invariants and A R, R is the diaonal matrix A R, R 4 R 1 4 R R 4 R v c 0. 0 v c Numerous studies concernin boundary controllability for quasilinear hyperbolic system were established [1], [2], [9], [6], [5], [7], [8]. Guat and Leuerin [4] were able to prove that one can control lobally the flow of fluid from one steady subcritical state to another see also []. Particularly, [4] considered fluid flow throuh a frictionless horizontal channel that is described by the quasilinear hyperbolic form of the de St. Venant equations R R A R, R R x 0. 5 R Recently, Mendoza, Valdez, and Arceo [10] extended the main result in [4]. They were able to show that the de St. Venant system 5 can be steered from a iven unsteady subcritical state to another in finite time by applyin nonlinear boundary controls in such a way that the solution is continuously differentiable. Likewise, the derivatives of the solution and boundary controls are also bounded. In this paper, we consider the numerical implementation as a verification for the existence of a lobal boundary control for the de St. Venant systems 5 from a iven unsteady subcritical state to another in finite time. II. RESULTS In this section we state the followin results proved in [10] concernin the de St. Venant equations steered between unsteady subcritical states. For the definitions and notations please refer to [10]. We start by statin the main theorem as follows:

2 Theorem 1. Consider the de St. Venant system 5 with boundary conditions of the form R 0, t 1 t and R L, t 2 t. Let a nonempty compact rectanular set Ω [a, b ] [a, b ] R 2 be iven such that for all d, e Ω, we have 4 d 1 4 e > 0 and 1 4 d 4e < 0, i.e., Ω contains only subcritical states. Given an ɛ > 0, let Ω ɛ [a ɛ 2, b ɛ 2 ] [a ɛ 2, b ɛ 2 ]. Then we can find boundary controls 1, 2 that steer the de St. Venant system with boundary conditions R 0, t 1 t and R L, t 2 t in finite time T from any unsteady initial state Φ C 1 [0, L], R 2, Φ[0, L] C 1 Ω ɛ to an unsteady state Ψ C 1 [0, L], Ω in such a way that the correspondin solution is continuously differentiable. Moreover, this can be achieved in such a way that the absolute values of the derivatives of the solution and of the controls 1, 2 remain smaller than any iven upper bound. For the outline of proof of Theorem 1, two lemmas are used. Lemma 1 is concerned with applyin boundary controls which steer the de St. Venant system from unsteady subcritical initial state Φ which is continuously differentiable Ω ɛ -valued function on [0, L] to a constant terminal subcritical state Φ 1 Ω in finite time T 1 provided the norm of the difference of the initial and final states is bounded by α > 0. Moreover, durin this transition the correspondin solution is continuously differentiable. Furthermore, the absolute values of the derivatives of the solution and the controls are bounded. On the other hand, Lemma 2 is concerned with applyin boundary controls which steer the de St. Venant system from a steady subcritical state Φ 1 Ω to an unsteady subcritical state Ψ which is continuously differentiable Ω- valued function on [0, L] in finite time T 2. Also, durin this transition the solution is continuously differentiable. Likewise, the absolute values of the derivatives of the solution and the controls are bounded. Thus the theorem is proved by combinin the two lemmas. III. NUMERICAL SOLUTION In this section we utilize the method of characteristics in determinin the numerical solution of the de St. Venant system steered from a iven state to another unsteady to steady or vice versa in finite time. Numerically, we consider only the extended lenth of the open channel L e L 2δ, where L is the lenth of the channel and δ > 0. The extended lenth of the open channel is subdivided into N x intervals for N N. Initially at time t 0, for every point in the extended lenth of the open channel, velocity, heiht and celerity which are all continuously differentiable functions are iven initial values. Likewise, the followin constants are used, ravitational acceleration 9.81, slope of the channel θ 0 and coefficient of friction s f 0. The solution for a particular point P in the advanced time t t 1 from a previous time t is numerically the intersection v c and v c emanatin from points Q and R at time t. The reion bounded by the forward and backward characteristics from points Q and R throuh point P is referred to as the domain of dependence of point P. When determinin the value of the solution at a particular point P, the information comin from the domain of dependence of point P is sufficient to achieve stability of the forward characteristic dv2c backward characteristic dv 2c and accuracy of the numerical solution at point P. Thus stability and accuracy of the numerical solution depend on the values of t and x. The forward characteristic from point Q to point P has slope dx 1 vc and the backward characteristic from point R to point P has slope dx 1 v c. To remain within the domain of dependence of point P, dx > v c must be satisfied. This stability criteria is referred to as the Courant-Friedrichs-Lewy CFL condition. Hence, to ensure stability of numerical calculation for every point in the extended lenth of the channel for a particular period of time, we have to choose the time step t 1 such that x t 1 t maxv, where v ic i i is the initial velocity, c i is the initial celerity at initial time t 0 for i 1,, N 1. We now formulate an alorithm in steerin the de St. Venant system from a iven state to another state in finite time T by the followin pseudocode below. Step 1. Set the values of the followin: time T lenth L δ > 0 L e L 2δ N N x Le N x [0 : N] x 9.81 slope of the open channel θ coefficient of friction s f initial states: velocity v i, heiht h i, celerity c i x t T t 1 maxv ic i N s.t. t 1 < t CFL condition T T [0 : t 1 T N ] t 1. Step 2. Define the initial condition on [ δ, L δ]. for n 1,..., N 1 v i x n, 0 c i x n, 0 h i x n, 0. Let V [v i x 1, 0 v i x N1, 0] C [c i x 1, 0 c i x N1, 0] H [h i x 1, 0 h i x N1, 0]. Step. For n 1,..., N 1 determine vx n, t 1 and cx n, t 1 from a previous time t. for t 1,..., T N for n 1,..., N set the values of v m, c m as v m x n, t vixn,tvixn1,t 2 2 c m x n, t cixn,tcixn1,t for n 2,..., N vx n, t v mx n 1, t 1 2 v mx n, t c m x n 1, t c m x n, t t θ s f cx n, t v mx n 1, t 1 4 v mx n, t 1 2 c mx n 1, t 1 2 c mx n, t hx n, t 1 c2 x n,t1 for n 1 vx 1, t 1 v m x 1, t cx 1, t 1 c m x 1, t

3 hx 1, t 1 c2 x 1,t1 for n N 1 vx N1, t 1 v m x N, t cx N1, t 1 c m x N, t hx N1, t 1 c2 x N1,t1 V x 1 : x N1, t 1 [ V vx 1 : x N1, t 1 T ] Cx 1 : x N1, t 1 [ C cx 1 : x N1, t 1 T ] Hx 1 : x N1, t 1 [ H hx 1 : x N1, t 1 T ] v i x 1 : x N1, t vx 1 : x N1, t 1 c i x 1 : x N1, t cx 1 : x N1, t 1. Step 4. Plot the raph of the heiht, velocity or celerity of the open channel. IV. NUMERICAL EXAMPLES Executin our code in Scilab version 4.1.1, we establish th followin numerical examples. We first numerically verify the existence of boundary controls that steer the de St. Venant system from a iven unsteady subcritical state to a steady subcritical state. We define an initial state Φ which is a continuously differentiable function satisfyin conditions of Theorem 1 in [10]. Example 1. We want the unsteady subcritical state Φx Φ 1 x, Φ 2 x Ω ɛ, where Φ 1 x vx 2cx and Φ 2 x vx 2cx steered to a constant subcritical state d 2, e 2 Ω. To do this, first let δ > 0 be iven. To work with the Cauchy problem the initial state Φ must be defined in an infinite domain. Thus we define explicitly the function Φ Φ 1 x, Φ 2 x as follows: d 2, x, δ, d 2 f 1 x, x [ δ, 0, Φ 1 x Φ 1 x, x [0, L], d 2 f 1 L δ x, x L, L δ], d 2, x L δ, ], and e 2, x, δ, e 2 f 2 x, x [ δ, 0, Φ 2 x Φ 2 x, x [0, L], e 2 f 2 L δ x, x L, L δ], e 2, x L δ, ], where with 2x f i x α i x δ x4 δ 4 for i {1, 2}, α 1 x Φ 1 x d 2 and α 2 x Φ 2 x e 2. Initially for the heiht of the fluid we define a nonconstant C 1 -function as 0.5, x, δ, β 1 x, x [ δ, 0, ĥ i x β 2 x, x [0, L], β x, x L, L δ], 0.5, x L δ, ], where 2 γ x δ 5 x 52 2 x , β 1 x β 2 x cos2πx, β x γ x δ 5 25 x x , with γ 1 x cos2πx , γ 2 x cos2π25 x , and we define the initial velocity as V i x cosπx, for all x R. Numerically, we only consider the strip [ δ, L δ] [0, T 1 ], where L 10, δ 5, and T 1 > 0. The interval [0, L] corresponds to the space interval [10, 20], the interval [ δ, 0] corresponds to [5, 10] and the interval [L, L δ] to [20, 25]. Fiures 1, 2, correspond to the initial heiht, initial velocity, and initial celerity of the fluid, which are all continuously differentiable functions, respectively. Note that we satisfy the conditions of Theorem 1 in [10], i.e., we consider only the subcritical state of flow. Fi. 4 shows the numerical existence of nonlinear boundary controls that steer the de St. Venant system from an initial velocity and an initial heiht rihtmost defined above towards a constant heiht of 0.5 leftmost with a constant velocity of 1.98 in finite time T 1. Second, we verify numerically the existence of boundary controls that steer the de St. Venant system from a iven steady subcritical state to an unsteady subcritical state in finite time T 2. Example 2. Now, consider the steady subcritical state d 2, e 2 Ω. To direct it towards an unsteady subcritical state Ψx Ψ 1 x, Ψ 2 x Ω, where Ψ 1 x vx 2cx, and Ψ 2 x vx 2cx, we work with the Cauchy problem, i.e., we must extend the initial state in an infinite domain. Define explicitly the function Ψx Ψ1 x, Ψ 2 x as follows: Ψ 11 x, x, 0, Ψ 1 x d 2, x [0, L], Ψ 12 x, x L, ], and Ψ 21 x, x, 0, Ψ 2 x e 2, x [0, L], Ψ 22 x, x L,, where Ψ ij x are iven for i, j 1, 2. Initially for the heiht of the fluid we define a C 1 -function 0.52 cosπx, x, 0, ĥ i x 0.5, x [0, L], 0.52 cosπx, x L,, and we define the initial velocity V i x 1.96, for all x R. Aain, numerically, we only consider the strip [ δ, L δ] [0, T 2 ], where L 10, δ 15, and T 2 > 0. The interval

4 [0, L] corresponds to the space interval [15, 25], the interval [ δ, 0] corresponds to [0, 15] and the interval [L, L δ] to [25, 40]. Fiures 5, 6, 7 correspond to the initial heiht, initial velocity, initial celerity of the fluid, which are all continuously differentiable functions, respectively. Note also that we satisfy the conditions of Theorem 1 in [10], i.e., we consider only the subcritical state of flow. Fi. 8 shows the numerical existence of boundary controls that steer the de St. Venant system from a constant heiht of 0.5 rihtmost with a constant velocity of 1.98 towards a varyin heiht leftmost in finite time T 2. Combinin the two numerical examples above, we have shown numerically the existence of boundary controls that steer the de St. Venant system from a iven unsteady state Φ C 1 [0, L], Ω ɛ towards another unsteady state Ψ [0, L], Ω in finite time T T 1 T 2. Fi. 4. Fluid heiht Unsteady subcritical state to steady state. Fi. 5. Fluid initial heiht. Fi. 1. Fluid initial unsteady heiht. Fi. 6. Fluid initial velocity. Fi. 2. Fluid initial unsteady velocity. Fi. 7. Fluid initial celerity. Fi.. Fluid initial celerity.

Fi. 8. Fluid heiht Steady state to unsteady subcritical state. REFERENCES [1] M. Cirinà, Boundary Controllability of Nonlinear Hyperbolic Systems, SIAM J. Control, vol.7, May 1969, pp.198 212. [2] M.

5 Fi. 8. Fluid heiht Steady state to unsteady subcritical state. REFERENCES [1] M. Cirinà, Boundary Controllability of Nonlinear Hyperbolic Systems, SIAM J. Control, vol.7, May 1969, pp [2] M. Cirinà, Nonlinear Hyperbolic Problems with Solutions on Preassined Sets, Michian Math J., 1970, vol.17, pp [] M. Guat, Boundary Controllability Between Sub-and Supercritical Flow, SIAM J. Control Optim., 200, vol.42, pp [4] M. Guat, G. Leuerin, Global Boundary Controllability of the de St. Venant Equations Between Steady States, Ann.I.H. Poincare, AN 20, 200, pp [5] M. Guat, G. Leuerin, Global Boundary Controllability of the de St. Venant System for Sloped Canals with Friction, Ann. I. H. Poincare, AN 26, 2009, pp [6] T. Li, Exact Boundary Controllability of Unsteady Flows in a Network of Open Canals, Math.Nachr., vol.278, 2005, pp [7] T. Li, B. Rao, Exact Boundary Controllability of Unsteady Flows in a Tree-Like Network of Open Canals, Meth. Appl. Anal., vol.ii, 2004, pp [8] T. Li, B. Rao, and Y. Jin, Semi-Global C 1 Solution and Exact Boundary Controllability for Reducible Quasilinear Hyperbolic System, Math. Modell. Num. Anal., vol.4, 2000, pp [9] T. Li, Z. Wan, Global Exact Boundary Controllability for First Order Quasilinear Hyperbolic Systems of Diaonal Form, Int.J. Dynamical Systems and Differential Equations, vol.1, 2007, pp [10] A. Mendoza, A.R. Valdez, and C. Arceo, Global boundary controllability of the de St. Venant equations between unsteady subcritical states, Proc. Int. Conf. on Computer and Software Mod. 2010, pp

Index. Chromatography, 3 Condition of C 0 compatibility, 8, 9 Condition of C 1 compatibility, 8, 9, 33, 36, 38, 39, 41, 43, 50, 53, 56, 58, 59, 62

Index. Chromatography, 3 Condition of C 0 compatibility, 8, 9 Condition of C 1 compatibility, 8, 9, 33, 36, 38, 39, 41, 43, 50, 53, 56, 58, 59, 62 References 1. Bressan, A., Čanić, S., Garavello, M., Herty, M., Piccoli, B.: Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1, 47 111 (2014) 2. Coron, J.-M., Wang, Z.: Controllability