Optimal profiles in variable speed flows

Optimal profiles in v ariable sp eed flo ws Gianluca Argen tini [0,1]Bending - Italy 01b ending@gmail.com gianluca.argen t ini@ g mail.com F ebruar y 2009 Abstract Where a 2D prob lem of optimal profile in v ariable sp eed flow is resolv ed in a class of con vex Bezier curv es, u sing symbolic and n umer- ical computations. Keyw ords : Newton pressur e la w, Bezier curv es, optimal shap e de- sign, computational engineering. W e study a particular problem on searc hing the optimal 2D shap e o r pro - file of a cartesian ob ject immersed in a v a r ia ble sp eed flow a nd sub jected to some constrain ts on the b oundary . Let b e { x, y } a cartesian system, and y = f ( x ) a function, defined for semplicit y on the unit interv al [0 , 1], whose graph is the profile which w e w an t to optimize. The function f is sub jected to the f ollo wing constrain ts, arising from engineering requiremen ts: c1. f (0) = 0 c2. f (1) = 0 c3. f is con v ex in (0 , 1) 1 Assume t ha t the fr ees t r eam flow is a ve ctor field par a llel to y -axis with a sp eed distribution according to a function v = v ( x ) dep ending on x -v ariable. Also, assume that the flow pressure acting on the ob ject profile is determined b y Newton’s sin-squared law ([2]), tha t is, b eing ρ t he fluid constant densit y and α the angle b et w een flow direction and tangent to profile, p ( x ) = ρv 2 ( x ) sin 2 α = ρv 2 ( x ) 1 1 + f ′ 2 ( x ) (1) The lo cal force acting on a n elemen t profile of x -extens ion dx is p ( x ) d x , and this is the expression of the lo cal resis tance offered t o the flow b y the bo dy . The total f orce on the profile is therefore F = ρ Z 1 0 v 2 ( x ) 1 + f ′ 2 ( x ) dx (2) The purpo se of the mathematical analysis is the minimization of this func- tional in a suitable class of functions where one hop es to find the solution function f ( x ). In g eneral, usual metho ds o f calculus of v ariations are applied, but one can easily test that techn ical difficulties arise in simple cases to o. F or example, let b e v ( x ) = ax . The Euler-Lagrang e equation ( see [3]) a ss o ciated to the minimization problem is than 0 = d dx ∂ F ∂ f ′ = − 2 ρa 2 d dx x 2 f ′ (1 + f ′ 2 ) 2 (3) from whic h a fo ur-degree algebraic equations in the v ariable f ′ arises, with c arbitrary constan t: x 2 f ′ + c  1 + f ′ 2  2 = 0 (4) Suc h equation is not simple to manage and not so useful. It is more istruc- tiv e, and useful fo r concrete applications, to consider a class of functions f in t the domain of Bezier curv es. Consider the three con t r o l p oin ts { P i , i = 0 , 1 , 2 } = { 0 , 0 } , { a, 1 } , { 1 , 0 } , where 0 ≤ a ≤ 1. The associated Bezier curv e (se e [1]) is b ( t ) = ( b 1 ( t ) , b 2 ( t )) = 2 X i =0 P i  2 i  t i (1 − t ) 2 − i = { 2 at + (1 − 2 a ) t 2 , 2 t − 2 t 2 } (5) 2 Figure 1: Some Bezier curves in the c ase a = 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 , and r elative c ontr ol p oints. Note that the para metric curve b satisfies the conditions c1. and c2., respec- tiv ely for t = 0 a nd t = 1, and c3. (see Fig.(1)). Also, x ′ ( t ) = 2 a + 2(1 − 2 a ) t , and some us ual algebraic c omputations sho w that x ′ ( t ) 6 = 0 if 1 2 < a < 1. Under this assumption, if f = f ( x ) is a cartesian represen tation of b , w e hav e f ′ ( x ) = d f dx = db 2 dt dt dx = db 2 dt  db 1 dt  − 1 = 1 − 2 t a + (1 − 2 a ) t (6) and dx = b ′ 1 dt , so t hat the functional (2) to minimize b ecomes F = F ( a ) = ρ Z 1 0 v 2 ( b 1 ) b ′ 3 1 b ′ 2 1 + b ′ 2 2 dt (7) No w the minimization problem is finding the v alue of the v ariable a , if exists, for whic h previous functional has its minim um v alue. Consider, for example, the case of a freestream sp eed distribution v ( x ) = − 5 x 3 ; this situation has b een pro p osed b y a customer, as ve lo cit y field a t oulet o f a ven tilating system and g oing to wards a conv ex b o dy . Along the profile of the Bezier curve (5) the sp eed distribution has expression v ( x ( t )) = − 5 (2 at + (1 − 2 a ) t 2 ) 3 and the functional F b ecomes F = F ( a ) = ρ Z 1 0 50 t 6 ( a + t − 2 at ) 3 (2 a + t − 2 at ) 6 ( a + t − 2 at ) 2 + (1 − 2 t ) 2 dt (8) 3 W e note that previous is an in tegral of a rational function and it is sym b ol- ically computable, but for our purp ose we can only compute it nume rically for a suffi cien t set of v alues of the parameter a , equally spaced in the inter- v al [0 , 1], and v erify the existence of a lo cal minim um for F ( a ). Figure (2) sho ws the evidence of suc h a minim um for a v alue 0 . 6 < a 0 < 0 . 8. An exact computation, by usu al n umerical tec hniques, giv es the v alue a 0 = 0 . 682564 (our computation: use o f FindMinim um , by Mathematic a v.7, (see [4])). Figure 2: Gr aph of F ( a ) in the c ase v ( x ) = − 5 x 3 . The problem of finding the optimal profile in the suitable class of Bezier curv es is so resolv ed in this concrete case. Figure 3: The optima l pr ofile. The p oint of maximum is in th e r e gion wh e r e the flow sp e e d values ar e gr e ater. 4 References [1] R.H.Ba r tels, J.C.Beatt y and B.A.Barsky , A n Intr o duction to S plines for Use in C omputer Gr aphics and Ge ometric Mo del ling , San F rancisco, Mor- gan Kaufmann, 1 998 [2] T.Lachand-Rob ert and M.A.P eletier, Newton’s Problem of the Bo dy of Minimal R esis t a nce in t he Class o f Con ve x Deve lopable F unc tions, Math- ematische Nachrichten , 226 , 153-17 6 , 2001 [3] L.Leb edev and M.Cloud, The Calculus o f V ariations an d F untional Anal- ysis with Optimal Contr ol a nd Applic ations in Me chanics , W orld Scien- tific, 2003 [4] S.W ag on, Ho w quic kly does water co ol?, Mathema tic a in Educ ation and R ese ar ch , 10 , 3, 2005 Gianluca Argen tini , mathematician, w orks on the field of fluid dynamics, of acoustical noise re duction and optimization of shap es for b odies moving inside fluid flows. He has found [0,1]Bending , a Design S tudio in Italy dedicated to computational engineering for scientific and i ndustrial applications. 5