Update on Progress with Component 4, Local Scale Trajectory Model John Wilson, Aug/05 intended capabilities review of where things stood Oct/04 Eugenes analytic integration new test simulations water-channel CUBES summary Intended capabilities of urbanLS.for Compute the (forward) transport and mixing of puffs/plumes of hazard gas from one or more steady or transient sources within an urban-type flow Function within a master program that supplies grid, flow field, source positions and strengths. Compute trajectories from source(s) out to path termination or path handover to another model Trajectory-simulation (ie. Lagrangian) type model (as opposed to Eulerian diffusion model) in the interests of: range-independent validity; optimal use of available wind statistics; simplicity and flexibility; state-of- art approach Crucial that program be robust and its predictions believable - thus show it is reliable (relative to reasonable expectations) by demonstrating its accuracy over a range of challenging test cases Eulerian Mass is conserved... Lagrangian Things move... ... during advection and diffusion between control volumes... memoryrandomforcing i-1, j-1 i+1, j+1 t+dt t + closure Status of urbanLS as of Oct/04 Status of urbanLS as of Oct/04 x y Row 38.5 mm 62.9 mm 11.8 mm Obstacle height H = 12.4 mm 59.4 mm A B C D E F G H II J K L M N O P tracer released at L Scale: 1:205 Geometry of MUST Array Simulation 3D, C 0 =4.8, dt/T L = 0.1, 9 reps of N P =50K. Inverse of the stress tensor ( R -1 ) regularized using an adaptive limit; reciprocal of condition number ( = 1/N c ) limited to = 0.1 or = 0.01 Detector volumes 2 x 2 x 0.4 cm 3 z/H=0.75 x/H=16 Eulerian Lagrangian LS simulation for steady source (vs. obs. and vs.Eulerian) data But: latest simulation using urbanLS2.for (all terms retained) overestimates peak conc. by about 30% LagrangianEulerian Simulation for transient source (vs. obs.) Developments since last meeting Oct/04 Developments since last meeting Oct/04 refinements to several subroutines, eg. hitbuilding which detects a building has been penetrated, and bounces the particle off the appropriate wall begin tests with new flow/tracer fields (water channel CUBES). Re-encounter problem of rogue trajectories introduce and test Eugenes semi-analytic time integration (resulting in many aspects of earlier program, eg. inverting R, becoming obsolete) establish that for this flow field urbanLS gives C(y) transects that are too narrow and peaky. Played with a superposed (independent) low-freq crosswind velocity component (at risk of over-tuning) however the Eulerian simulations also give C(y) transects that are too narrow and peaky. Hypothesized cause: urbanSTREAM underpredicting turbulent kinetic energy (in the aligned array of cubes, in particular). Eugene provided revised RANS flow field and an alternative URANS flow field. Outcome not categorically better Will demo ditto Rogue trajectories Rogue trajectories Luhar & Britter (1989) imposed extraneous numerical constraint on magnitude of the velocity pdf to prevent the occurrence of unrealistic velocities (1d model of convective b/l) Naslund et al. (1994) reported the Thomson LS model applied to compute trajectories around a building becomes unstable unless the stress tensor satisfies realizability ( G a standardized Gaussian random variate) velocity increment conditional mean acceleration Thomson3D-G Covariance retained, but inverse stress tensor R -1 regularized (enforcing condition number N c 100 ). But to eliminate the rogue trajectories, need N c 10 MUST (17 April/05) Condition number N c = 3 / 1 is the ratio of largest to smallest eigenvalues of the Reynolds stress tensor. Eugene had reasoned that when urbanLS evaluates acceleration term with factor R -1 u, perturbations in velocity u are amplified by the factor N c - so, limit this factor ? example Covariances set to zero (therefore no regularization). No velocity resets. Notice a couple of paths through building space (1st downwind row) CUBES (22 Mar/05) example proving that: proving that: diagonalizing R or regularizing R -1 may help, but doesnt universally solve the problem of rogue trajectories (nor is reducing timestep the solution) Eugenes hypothesis: Two causes for the observed instabilities potential presence of unstable modes intrinsic to the dynamical system (these modes sometimes excited in the complex flow fields within the urban arrays) the simple Euler scheme used to integrate the Langevin equations not sufficient to keep round-off errors under control, because these equations constitute a stiff system (loosely, combine widely differing rate constants) First cause exacerbates the second, but even if the first cause were eliminated the second cause (potentially) remains Semi-analytic time integration Semi-analytic time integration 1) Split the velocity increment into linear and non-linear (quadratic) contributions The linear part (well drop the (1) notation) can be written where Since L is not diagonal, correctly evaluating du 1 involves the history during the step of u 1, u 2, u 3, ie. we have interconnected equations. Dimensionally and operationally, L is an inverse timescale (matrix) for relaxation of velocity fluctuiations towards zero; when ratio 3 / 1 of largest- to-smallest magnitudes of (real parts of) eigenvalues of L is large, system is stiff Assume L has 3 real eigenvalues i ; use the corresponding eigenvectors as a basis for the velocity vector. Thus decompose L as where and the columns of S are the eigenvectors L and its associated eigenvalues (diagonal entries of ) and eigenvectors (columns of matrix S) are properties of the flow field, and vary from gridpoint to gridpoint. They are computed by urbanSTREAM and provided to urbanLS2, along with other needed derivative properties of the flow. 2) solve the linear part exactly by the following steps: In the eigenframe of L, the velocity vector is and the (linearized) Langevin equations, previously coupled, separate into three independent equations If U i,0 rot is the value at the beginning of the step, the solution for the final value is where i are correlated Gaussian random numbers and 3) the non-linear contribution is of which a part can be split off and handled exactly, (no summation over i ) with solution intermediate value resulting from update using linear terms The other quadratic terms are handled by elementary Euler integration MUST simulation: with the off-diagonal terms in u j u k MUST simulation: with the off-diagonal terms in u j u k (10aug05c) Nb! However concentration fields differ only very slightly dt/T L =0.01 dt/T L =0.1 (10aug05) Still a few rogues? MUST simulation: with and without the off-diagonal terms in u j u k MUST simulation: with and without the off-diagonal terms in u j u k In both cases: dt/T L =0.1 (10aug05) (10aug05b) Nb! However concentration fields differ only very slightly without: no rogues with proving that: proving that: the problem of rogue trajectories subtly hinges on the adequacy of the integration scheme and the dynamics of the generalized Langevin equation the new scheme is guaranteed to eliminate rogue trajectories only if the off-diagonal terms of the non-linear contribution are eliminated (for the new scheme offers no alternative but the Euler integral, for this component) refinement of the timestep will sometimes suffice to suppress unrealistic velocities away from plume margines, mean concentration field is not necessarily greatly impacted by the occurrence of rogue trajectories Performance in highly disturbed flow (water channel) Coanda Research and Development Corporation Square CUBE array run by urbanLS2 15apr05b, dt/T L =0.1 Lagrangian simulations too narrow and peaky MUST array run by urbanLS2 17 & 20April/05 (need dt/T L =0.01 or rogues) 14 Jun05a original flow field (decomposeL) 14 Jun05b, new flow field decomposeL_square_RANS 14 Jun05c, new flow field decomposeL_square_URANS Aligned CUBE array, source on top of building. Using analytical integration of Langevin eqn over dt (=0.1 T L ), with all terms retained proving that: proving that: the particulars of the flow field also bear on the occurrence of rogue trajectories Aligned CUBE array, source on top of building. Using analytical integration of Langevin eqn over dt (=0.1 T L ), with all terms retained/off-diagonals dropped 14Jun05c, URANS, all terms 11aug05, URANS, drop off-diag-non-linear terms but, very little alteration of mean C field RANS URANS (with/without rogues) RANS URANS (with/without rogues) Conclusions analytic integration used in urbanLS2 eliminates most rogue trajectories even in pathologically complex flowfield (all? - if drop off- diag- non-linear terms) earliest Lagrangian simulations of MUST steady source were of slightly better accuracy than Eulerian Lagrangian simulations of transient MUST source were markedly superior to Eulerian as regards peak concentration urbanLS2 has been tested adequately against lab data; ready to be tested against field data in CUBE array (and to lesser extend, MUST array) both Lagrangian and Eulerian simulations of C(y) too narrrow and peaky. This is in part (if not in total) due to the provided flow field it is anyway unrealistic to expect to agree within 50% everywhere (ie. plume centreline and plume margins) in these complex flows not likely any other model could do better; nor likely urbanLS can be categorically improved on timescale of this (CRTI) project function as sub-program, called by a master-program read in a 3D grid whose origin is defined by master read in flow statistics (U,V,W, R ij, ) for each gridpoint (I,J,K) read in field identifying which cells (I,J,K) are building space read in locations, relative strengths of (multiple) steady and/or transient sources read in locations and geometries of detectors perform checks on flow field (? - this step now handled in urbanSTREAM) compute an ensemble of paths from each source handover path to another trajectory program on whole-city scale? compute hazard concentration at detectors Intended capabilities of urbanLS.for (ctd)