CFD with OpenSource software

Transcription

1 CFD with OpenSource software The Harmonic Balance Method in foam-extend Gregor Cvijetić Supervisor: Hrvoje Jasak Faculty of Mechanical Engineering and Naval Architecture University of Zagreb September 27, 2016 September 27, 2016

2 Table of Contents Introduction Mathematical Model Scalar Transport Navier Stokes Equations Validation of Harmonic Balance ERCOFTAC Centrifugal Pump OtaBm2 Centrifugal Pump Conclusion September 27, 2016

3 Introduction Many modern CFD problems involve temporally periodic flows: wave like phenomena, flows induced by periodic boundary condition, flows with periodically moving objects wing oscillations, moving valves, flutter, etc. Some of them are quite complex - turbomachinery. 3/33

4 Common problems: different specialised steady state methods available, but usually transient simulation is needed for resolving transient effects high CPU times for whole simulation in order to neglect simulation start unsteadiness, a number of periods have to be run - one period is not enough in specific cases up to 100 periods were needed to reach periodic steady state 4/33

5 Harmonic Balance Method method for temporally periodic problems where the fundamental frequency is known in advance it is assumed that each primitive variable can be accurately represented by a Fourier series in time, using first n harmonics and mean value instead of running transient simulation, 2n + 1 steady state problems for 2n + 1 equally spaced time steps within one period are solved 5/33

6 time derivative term is transformed into source terms denoting coupling with other time steps method is computationally more efficient and faster than conventional transient methods (substituting transient simulation with a number of steady state problems) memory increase implemented in foam-extend 6/33

7 Harmonic Balance for Scalar Transport The derivation Starting point is a scalar transport equation in condensed form: Q t + R = 0. R stands for the convection and diffusion transport and source/sink terms of the transport equation: R = (uq) (γ Q) q v 7/33

8 Discrete Fourier series expansion of scalar variable Q with n harmonics: Q(t) = Q 0 + Q S1 sin(ωt) + Q C1 cos(ωt) + Q S2 sin(2ωt) + Q C2 cos(2ωt) Q Sn sin(nωt) + Q Cn cos(nωt), Expansion of R: R(t) = R 0 + R S1 sin(ωt) + R C1 cos(ωt) + R S2 sin(2ωt) + R C2 cos(2ωt) R Sn sin(nωt) + R Cn cos(nωt). 8/33

9 Grouped together, 2n + 1 equations are obtained: ωq C1 + R S1 = 0 2ωQ C2 + R S2 = 0 n for sine... nωq Cn + R Sn = 0 mean R 0 = 0 n for cosine ωq S1 + R C1 = 0 2ωQ S2 + R C2 = 0... nωq Sn + R Cn = 0, 9/33

10 Written in matrix form: ωaq + R = 0. Matrix A, Q and R are coefficients matrices: A = n n 0, Q = Q S1 Q S2 Q S3. Q Sn Q 0 Q C1 Q C2 Q C3. Q Cn, R = R S1 R S2 R S3. R Sn R 0 R C1 R C2 R C3. R Cn 10/33.

11 Using a DFT transformation matrix E: Q = EQ(t) and R = ER(t), it is possible to keep the equation in frequency domain but use variables in the time domain: ωaeq + ER = 0 11/33

12 Multiplying the equation by transformation matrix E 1 from the left, the final form of the Harmonic Balance transport equation in time domain is obtained! ORIGINAL EQUATION HB FORMULATION Q t + R = 0 ωe 1 AEQ + R = 0 Time-domain matrices Q and R contain variable in 2n + 1 time steps: Q t1 R t1 Q t2 R t2 Q = Q t3, R = R t3.., Time, s Scalar value Q t2n+1 R t2n+1 where t n stands for: t 1 = T 2n + 1, t 2 = 2T 2n + 1,..., t n = nt 2n /33

13 Developing the equation ωe 1 AEQ + R = 0, 2n + 1 equations are obtained: Q t1 : (uq t1 ) (γ Q t1 ) = 2ω ( Qt2 P 1 +Q t3 P Q t2n+1 2n+1 P 2n) Q t2 : (uq t2 ) (γ Q t2 ) = 2ω ( Qt2 P 1 +Q t3 P Q t2n+1 2n+1 P 2n). where (uq tj ) (γ Q tj ) = 2ω 2n+1 ( 2n i=1 P (i j) Q ti ) n P i = k sin(kωit 1 ), for i = {1,2n} k=1 13/33

14 Harmonic Balance for Navier Stokes Equations Using velocity matrix u instead of Q yields Harmonic Balance momentum equation: ( 2n ) (u tj u tj ) (γ u tj ) = 2ω P 2n + 1 (i j) u ti General form of scalar transport equation becomes: (u tj Q tj ) (γ Q tj ) = 2ω 2n + 1 i=1 ( 2n i=1 P (i j) Q ti ) Continuity equation remains the same, with u replaced with its discrete counterpart u tj : u tj = 0. 14/33

15 Example: Equation Set for 1 Harmonic E 1 AE = 2 2n P 1 P 2 P 1 0 P 1 P 2 P 1 0 ωe 1 AE ˆQ + ˆR = 2ω 3 0 P 1 P 2 P 1 0 P 1 P 2 P 1 0 Q t1 Q t2 Q t3 + R t1 R t2 R t3, 15/33

16 Q t1 : (uq t1 ) (γ Q t1 ) = 2ω 3 (Q t 2 P 1 + Q t3 P 2 ), Q t2 : (uq t2 ) (γ Q t2 ) = 2ω 3 (Q t 1 ( P 1 ) + Q t3 P 1 ), Q t3 : (uq t3 ) (γ Q t3 ) = 2ω 3 (Q t 1 ( P 2 ) + Q t2 ( P 1 )), Scalar value -P 1 = P 1 = Time, s P 1 = , P 2 = Calculated time instance O 16/33

17 In case of three harmonics: -P 2 = Scalar value -P 3 = P 1 = Calculated time instance O P 1 = P 2 = P 3 =1.795 Time, s P 1 = , P 2 = , P 3 = 1.795, P 4 = 1.795, P 5 = , P 6 = /33

18 Validation of Harmonic Balance Turbomachinery test cases: ERCOFTAC Centrifugal Pump, 2D OtaBm2 Pump, 3D Two-phase test cases: Regular Wave Propagation, 2D DTMB Ship on Waves, 3D 18/33

19 ERCOFTAC Centrifugal Pump - radial inlet velocity: 11.4 m/s - rotational speed: 2000 rpm - k-ε turbulence model - multiple-frequency approach f rotor = rpm/60 f stator = f rotor n rotor blades - no time interpolation on the rotor-stator interface! 19/33

20 Pump global parameters comparison: Transient HB, 1h error, % HB, 2h error, % MRF error, % Efficiency t = T 3 Head Torque Efficiency t = 2T 3 Head Torque Efficiency t = T Head Torque /33

21 Local transient effects comparison, velocity field: Transient simulation. HB with 1 harmonic. HB with 2 harmonics. MRF simulation. Table of Contents Introduction Mathematical Model Results Conclusion 21/33 September 27, 2016

22 Rotor blade pressure contours at 1/3T, 2/3T and T time instants: 0 0 Pressure, Pa Transient MRF HB, 1h HB, 2h Pressure, Pa Transient MRF HB, 1h HB, 2h Radius, m Radius, m a) t = T/3. b) t = 2T/3. 0 Pressure, Pa Transient MRF HB, 1h HB, 2h Radius, m c) t = T. 22/33

23 CPU time comparison: Transient 1 period 5 hours, 8 periods 40 hours MRF HB, 1h HB, 2h 3100 iterations, 20 minutes 3000 iterations, 52 minutes 2400 iterations, 78 minutes Transient simulation: time steps per rotation - 2 outer correctors - 4 inner correctors Harmonic Balance with 2 harmonics approximately 30 times faster than transient simulation. 23/33

24 OtaBm2 Centrifugal Pump - inlet velocity: m/s - rotational speed: 800 rpm - k-ε turbulence model Table of Contents Introduction Mathematical Model Results Conclusion 24/33 September 27, 2016

25 Pump global parameters comparison: Transient solver HB, 1h error, % MRF error, % Efficiency t = T 3 Head Torque Efficiency t = 2T 3 Head Torque Efficiency t = T Head Torque /33

26 Velocity field: Transient simulation. HB simulation, 1h. MRF simulation. Table of Contents Introduction Mathematical Model Results Conclusion 26/33 September 27, 2016

27 Rotor blade pressure contours at 1/3T, 2/3T and T time instants: Pressure, Pa Transient HB, 1h MRF Pressure, Pa Transient HB, 1h MRF Radius, m Radius, m t = T/3. t = 2T/ Pressure, Pa Transient HB, 1h MRF Radius, m t = T. 27/33

28 Regular Wave Propagation Periodic two phase surface flow Grid 1 Grid 2 Grid 3 η a,0, m η a,1, m N Iter 0 th and 1 st order amplitude convergence. 28/33

29 Zeroth and 1st order harmonic amplitudes of the free surface with iterative uncertainty obtained with different spectral resolution: No. Harmonics η a,0, m η a,1, m U I,0, % U I,1, % ɛ 0, m ɛ 1, % /33

30 DTMB Ship on Waves Transient simulation. Ship model parameters: L = 3.05 m, HB simulation. Froude number: F r = 0.28, U = 1.52 m/s, Wave parameters: H = m, T = 1.09 s, λ = 4.57 m, 2 harmonics are used in the HB simulation, 200 time steps per encounter wave period are used in the transient simulation. 30/33

31 Transient simulation comparison 16 F x,i, N th order 1st, order F z,i, N N Iter Convergence of longitudinal and vertical force harmonics. Item Transient Harmonic Balance ε, % F x,0, N F x,1, N F z,0, N F z,1, N /33

32 Conclusion Harmonic Balance method implemented in foam-extend validated for scalar transport and Navier Stokes equations, 2D and 3D computationally more efficient than conventional transient methods wide range of application Future work: implicit Harmonic Balance benchmarking for turbomachinery full multiple-frequency approach multistage turbomachinery periodic boundary conditions 32/33

33 Thank you for your attention. Questions? 33/33