Discussion of wind load

Journal of Wind Engineering and Industrial Aerodynamics, 46 & 47 (1993) 527-537

527

Elsevier

DISCUSSIONS OF WIND LOAD The Generalization and Simplification of Wind Loads and Implications for Computational Methods Author: A.G.DA VENPORT A . B A S K A R A N : C o u l d you please comment on whether computers will replace wind tunnels ?

Response:The relationships between computers and wind tunnels have to the present been very productive. In wind engineering, wind tunnels have created some extremely complicated turbulent flows around extremely complicated buildings in complicated urban landscapes. Only with computation of the statistical properties can we understand what's happening. I expect computers will expand the usefulness of wind tunnels, will increasingly become an alternative to wind tunnels, and carry out calculations wind tunnels cannot d o - such as compute flows in tornados. But I do not expect wind tunnels to be entirely replaced,

Numerical Simulation of Wind Geometries

Induced Pressures on Buildings of Various

Authors : T.STA THOPOULOS et eL A . K A R E E M : H o w do you plan to predict peak pressure coefficients based on your reported numerical scheme ?

Response:We are currently thinking of applying either the l a r g e - e d d y

simulation

technique or direct numerical simulation. It should be recalled, however, that at present no study has achieved such

computation of peak pressure coefficients and, in my

view, it will take a while until real progress occurs in this very important area.

R.P, SELVAM:You mentioned that simulation of flow separation on the top front roof is not possible using the k-e model. I have produced it, and

presented it at

the US Wind Eng. Conf. 1990. Also Dr.Gosman's presentation today has shown that you need a much finer mesh size to produce separation on the roof front.

528

Response Professor separation

: We h a v e c a r r i e d out v e r y d e t a i l e d s t u d i e s of grid r e f i n e m e n t s Murakami can

be

has

also s h o w n , d e p e n d i n g

achieved.

However,

there

on t h e are

still

boundary difficulties

r e p r e s e n t a t i o n of the p r e s s u r e fields, p a r t i c u l a r l y n e a r roof a p p l y i n g t h e t e c h n i q u e we

and, ::~s

c o n d i t i o n s , flow in

the

actual

edges a n d c o r n e r s w h e n

h a v e o u t l i n e d in t h i s paper.

J.D.HOLMES : If you c o m p u t e d p r e s s u r e s for full - scale b u i l d i n g s , w h a t d i f f e r e n c e s to y o u r c o m p u t a t i o n a l

a p p r o a c h would you m a k e ? W o u l d you

expect

significant

d i f f e r e n c e s in y o u r c o m p u t e d p r e s s u r e coefficients ?

Response:Attention

s h o u l d be paid to t h e c o m p u t a t i o n a l g r i d a n d p a r t i c u l a r l y to

its density. On the o t h e r hand, instead of the zonal t r e a t m e n t m e t h o d for the calculation of k a n d e n e a r t h e surfaces, t h e s t a n d a r d wall f u n c t i o n f o r m u l a t i o n for E s h o u l d be used, since sub-layer

of

V, fl

and

t h e f i r s t c o m p u t a t i o n a l g r i d l i n e would be out of the viscous

the flow. Hopefully, no s i g n i f i c a n t d i f f e r e n c e s in t h e c o m p u t e d p r e s s u r e

c o e f f i c i e n t s w o u l d be expected.

Predicting R.M.S. Pressures from Computed Velocities and Mean Pressures Author: D.A.PA TERSON A.LEONARD:Your d e r i v a t i o n does n o t seem to include t h e effect of v o r t i c i t y .

Response:It

is i n c l u d e d t h r o u g h t h e v a r i a t i o n of Co, a l t h o u g h p e r h a p s n o t

in an

o p t i m a l way.

A.MOCHIDA:The p r e s s u r e d i s t r i b u t i o n g i v e n b y y o u r c a l c u l a t i o n s h o w s a q u i t e d i f f e r e n t d i s t r i b u t i o n on t h e roof f r o m t h a t usually o b t a i n e d from the k-~ model, a n d the a g r e e m e n t w i t h m e a s u r e d d i s t r i b u t i o n s seems to be b e t t e r t h a n t h a t usually found. Did you i n t r o d u c e some special t r e a t m e n t r e g a r d i n g t h e wall b o u n d a r y c o n d i t i o n s ?

Response:Yes.

T h e s t a n d a r d l o g a r i t h m i c law does not a p p l y a t s e p a r a t i o n points.

A more a c c u r a t e b o u n d a r y condition was used beside t h e u p w i n d edges. In this b o u n d a r y condition, w h i c h I call the r e c i r c u l a t i o n p r o m o t e r , t h e l o n g i t u d i n a l v e l o c i t y is set to zero.

529

A Comparison of Computer and Wind - Tunnel Models of Turbulence around The Silsoe Structures Building Authors : P.J.RICHARDS et aL

W.RODI:Have you checked the numerical accuracy of your calculations? I believe the Phoenix code used employs an upwind differencing scheme which can introduce significant numerical diffusion.

Response : I have not checked the numerical accuracy. However, the numerical model results have been compared with f u l l - scale data, and at

least as far as the mean

pressure is concerned there is good correlation. This suggests to me that numerical diffusion is not too significant.

J.D.HOLMES:What are the typical relative contributions from the q u a s i - steady terms

(including the term associated with change of wind direction)

compared with

those associated with the kinetic energy ?

Response:Our

studies show that the pressure fluctuations are dominated by quasi

- s t e a d y effects, including changes in wind speed and direction. Local turbulent kinetic energy does not appear to be very significant.

N.HOSOYA :How do you allow for the size of the press taps installed, on the model or the f u l l - scale building, for computing the Co (RMS) in your numerical model ?

Response:No

corrections relating to tap size have been incorporated into

any of

the data presented.

T.STATHOPOULOS:Have you looked at the computation of the pressure coefficient on roof corner points? Previous studies have shown that even the mean values cannot be predicted accurately by the computer, for an oblique wind

direction, let alone the

RMS values.

Response:I

agree that this type of computer model tends to underestimate mean

pressure coefficients near roof corner points, and we must look for ways of overcoming this. However, it should be noted that the relationship between C, (RMS) and C, (mean) suggested in the paper is problem.

based on wind tunnel data, and is not affected by this

530

Computational and Experimental Roof Comer Pressures on The Texas Tech Building Authors : R.P.SELVAM et al.

J.D.HOLMES:Your p r e s s u r e d i s t r i b u t i o n s a p p e a r to show t h a t y o u r s i m u l a t i o n s d(~ not g e n e r a t e conical vortices b e h i n d the roof leading edge for a n oblique wind direction. Any comments ?

Response :

Yes, you are r i g h t . T h i s m a y be due to not h a v i n g e n o u g h cells. H a v i n g

more cells close to t h e roof c a n g e n e r a t e vortices.

J.H.FERZIGER:The difference b e t w e e n the results o b t a i n e d on the two grids indicates t h a t t h e e r r o r is r a t h e r l a r g e r a n d t h a t t h e correct r e s u l t s

are r a t h e r far from those

s h o w n for t h e fine grid.

Response:You

m a y b e r i g h t . T h i s n e e d s f u r t h e r s t u d y u s i n g f i n e r grids.

Numerical Simulation of Flowfleld around Texas Tech Building by Large Eddy Simulation Authors : A.MOCHIDA et al.

A.G.DAVENPORT:How did y o u r f l u c t u a t i n g p r e s s u r e m e a s u r e m e n t s c o m p a r e

with

Surry's measurements ?

Response

: D r . S u r r y ' s d a t a of f l u c t u a t i n g p r e s s u r e d i s t r i b u t i o n was n o t a v a i l a b l e to

us w h e n we c a r r i e d out this calculation. T h e r e f o r e in this our r e s u l t s of f l u c t u a t i n g p r e s s u r e

p a p e r we did not c o m p a r e

w i t h his m e a s u r e m e n t s . B u t it r e c e n t l y b e c a m e

possible to m a k e t h i s comparison, w h i c h was p u b l i s h e d in J. W i n d vol.38,

No.2-

measurements

3. In g e n e r a l , t h e

agreement

is good, e x c e p t for some

of

our

LES

Eng. Ind. Aerodyn.,

results

with

the

Surry

small differences. T h e l a r g e RMS p r e s s u r e

c o e f f i c i e n t s a t t h e f r o n t c o r n e r are s o m e w h a t u n d e r e s t i m a t e d in his m e a s u r e m e n t s , r e l a t i v e to our LES r e s u l t s a n d field m e a s u r e m e n t s . On t h e

other

RMS p r e s s u r e

by Surry

c o e f f i c i e n t s on t h e w i n d w a r d

agreement with field-measured

data

face m e a s u r e d

t h a n do t h e LES

results.

h a n d , v a l u e s of show b e t t e r

531

T.STATHOPOULOS:Have

you m a d e a n y a t t e m p t s to c o m p u t e p r e s s u r e s for

oblique

w i n d d i r e c t i o n s b y u s i n g LES ?

Reeponse:I h a v e c o n d u c t e d t h e c a l c u l a t i o n only for t h e case of 90 d e g r e e angle. I w o u l d like to t r y t h e c a l c u l a t i o n s for some cases of

wind

different wind angles

in t h e n e x t s t a g e of t h i s s t u d y .

B.BIENKIEWICZ:In

response

to

earlier

comments

on

the

need

for

benchmark

e x p e r i m e n t a l d a t a for c o m p u t a t i o n a l work, I would like to a n n o u n c e t h a t w i n d t u n n e l d a t a b a s e s for t h e T e x a s T e c h U n i v e r s i t y b u i l d i n g

( t e s t e d for a r a n g e of g e o m e t r i c a l

scales)

e x i s t a t Colorado S t a t e U n i v e r s i t y a n d can b e m a d e a v a i l a b l e in e l e c t r o n i c

and/or

printed format.

J.N.FERZIGER : It

is i m p o r t a n t to compare w i t h d a t a on planes o t h e r t h a n

the central

plane.

Large Eddy Simulation of Wind Flow around Dome Structures by The Finite Element Method Authors : T.OGAWA et al.

S.PARAMESWARAN:What

a b o u t t h e influence of grid size on t h e n u m e r i c a l r e s u l t s ?

A t l e a s t it is nice to k n o w w h a t h a p p e n s w h e n you r e d u c e t h e n u m b e r of e l e m e n t s to h a l f of its o r i g i n a l v a l u e

Response:We could

(3528 e l e m e n t s in t h i s case).

t r i e d some c a l c u l a t i o n s w i t h f e w e r e l e m e n t s , a n d on t h e w h o l e we

see t h e coincidences w i t h t h e p r e s e n t case of 3528 elements. But we could not

see t h e a p p e a r a n c e of t h e s e c o n d a r y vortices.

M.A.LESCHZINER : W h a t the time-

Response is s e c o n d -

FE f o r m u l a t i o n was used, a n d w h a t o r d e r of a c c u r a c y did

marching scheme have ?

: T h e G a l e r k i n f o r m was used. T h e a c c u r a c y of t h e t i m e - m a r c h i n g o r d e r in time.

scheme

532

Computation of Wind Flow over Topography Authors : D.A.PATERSON eLal

S.PARAMESWARAN:Is t h e g r i d s y s t e m used b o d y - f i t t e d

or C a r t e s i a n ? F r o m the

p i c t u r e of t h e grid, it is n o t clear t h a t the g r i d is a b o d y - f i t t e d

one.

Response : Body - fitted.

M.A.LESCHZINER : U p s t r e a m of t h e hill crest, no t u r b u l e n c e model is n e e d e d since t h e flow is g o v e r n e d b y i n v i s c i d processes. However, d o w n s t r e a m ,

y o u r coarse grid

a n d t u r b u l e n c e model will give t h e w r o n g b e h a v i o u r if t h e flow w a n t s to s e p a r a t e .

Response:Both the t u r b u l e n c e model a n d the coarse grid act to s u p p r e s s d o w n s t r e a m separation. T h i s is a p r o b l e m only for steep hills w i t h u p w i n d slopes of 0.6, and affects only t h e velocities in r e g i o n s w h e r e t h e

v e l o c i t y is small. T h e r e g i o n s of i n t e r e s t

are t h o s e w h e r e t h e v e l o c i t y is large, so t h i s is n o t a m a j o r p r o b l e m .

Analysis of Hyperbolic Cooling Towers for Wind Loads with ACMC and Semi - Loof Shell Elements Authors : KARISIDDAPPA et el.

J.D.HOLMES:I u n d e r s t a n d t h a t you h a v e studied t h e effects of t i m e - a v e r a g e d wind p r e s s u r e for cooling towers. Do you i n t e n d to s t u d y t h e e f f e c t of i n s t a n t a n e o u s or peak pressure distributions- these can take quite a

different form from the mean?

Some t e c h n i q u e s we developed in A u s t r a l i a for c i r c u l a r silos a n d t a n k s m a y be useful in t h i s r e g a r d .

Response: T h e w o r k p r e s e n t e d f o r m s p a r t of a n o n g o i n g p r o j e c t w h i c h i n v o l v e s a s t u d y of s e v e r a l issues r e l a t e d to t h e r e s p o n s e of cooling t o w e r s s u b j e c t e d to wind loading. It is also p r o p o s e d to s t u d y t h e

problem mentioned by the questioner. The

a u t h o r s will be g r a t e f u l if copies of r e p o r t s / p a p e r s d e v e l o p e d for silos a n d

t h a t are available on the t e c h n i q u e s

t a n k s can be s e n t to t h e m .

533

Computing The Statistical Stability of Integral Length Scale Measurements by Autoregressive Simulation Author: P.SCHRADER

A.KAREEM@:What

Response

is your AR order ?

The AR order used was 50 in the case of series GT, and 128 in the case

:

of series BT and BH. A very detailed discussion about AR order can be found in /13/.

A.KAREEM(~:Did Response

:

you consider the ARMA model to reduce the order ?

I did not try ARMA models, which might have been effective in reducing

computer time in the cases studied in my paper. However, on the one hand, additional computer time is often very inexpensive if one uses

one's own computer which is

there anyway and is normally not fully used at weekends and at n i g h t ; in the case of very high AR orders

(say 128 or even 256), a modern PC can generate 1000 or

2000 wind speed records during one night, each record corresponding to a 20 minutes' record in nature sampled at 5Hz. On the order hand, usage of AR models will often require much less work, which is very costly in modern industrialized countries; AR models can be learned faster than ARMA models by civil engineers;and their application and mathematical handling are simpler.

A.KAREEM(~):What

about the propagation of AR modelling uncertainty into your

uncertainty estimate of the integral scale ?

Response:Although

I am not quite sure that I have grasped the meaning of this

question fully, I am going to elaborate on it. First, I should like to state the following. In view of the fact that the

phenomena

studied in the paper are stochastic, as for the integral scale scatter estimate, I think that agreement between computer simulation and

experiment is quite good. From the

point of view of theoretical statistics, only one out of 12 simulated coefficients of variation was

too far away from the experimental one. Furthermore, the computer

simulation results for series BT and BH represent first try results, not

been

adjusted

by

trying

several

times

until

they

i.e., they have

closely approximate

the

experimental results; and the coefficients of variation obtained from AR models were

534

sometimes smaller and sometimes larger than

t h o s e o b t a i n e d from the w i n d t u n n e l

e x p e r i m e n t s . F o r each of the series GT, BT a n d BH, N -

800 s a m p l e s of each t y p e

of i n t e g r a l scale m e a s u r e m e n t were g e n e r a t e d w i t h t h e AR m o d e l s ; h e n c e , 2, these coefficients of v a r i a t i o n V are subject to a s t a t i s t i c a l u n c e r t a i n t y deviation)

ov-V/~/~

in

table

(a s t a n d a r d

w h i c h is a b o u t 2 . 5 % .

A detailed t h e o r e t i c a l i n v e s t i g a t i o n of the limitations a n d capabilities in c o n n e c t i o n w i t h m y p a p e r can be f o u n d in / 1 3 / .

of AR simulation

As for g e n e r a t i n g p r e s u m a b l y

s t a t i o n a r y wind t u n n e l w i n d speed f l u c t u a t i o n s w i t h a g i v e n s p e c t r a l density, the AN m o d e l s u s e d are s u b j e c t to t h r e e m a i n u n c e r t a i n t i e s : (1) T h e w i n d t u n n e l w i n d speed f l u c t u a t i o n s m a y be a bit n o n s t a t i o n a r y ,

while the

AR model is s t r i c t l y s t a t i o n a r y . (2) T h e AR s p e c t r a l d e n s i t y usually will n o t coincide fully w i t h t h e

spectral density

of t h e w i n d to be s i m u l a t e d . (3) T h e w i n d r e c o r d s g e n e r a t e d b y the AR model are Gaussian, while t h e m e a s u r e d wind records are not s t r i c t l y Gaussian. T h i s m e a n s t h a t t h e s t a t i s t i c a l m o m e n t s h i g h e r than second order

m u s t be e x p e c t e d to d i f f e r in t h e two cases, e v e n if t h e m e a n

wind speeds a n d the spectral densities coincide. F r o m the t h e o r y of stochastic processes it can be s h o w n t h a t , w h e n d e t e r m i n i n g the coefficients of variance, V, of wind speed variance

To, Tx, T~,~, T~v, in t h e g e n e r a l speed, E[/z'(t)./Z(t+rl)./z'(t+r2)./z'(t+r~)],

e s t i m a t e s , £, or of i n t e g r a l scale e s t i m a t e s ,

case, the f o u r t h - o r d e r s t a t i s t i c a l m o m e n t s of wind m a y e x e r t some influence.

In o u r wind t u n n e l , c o n d i t i o n s were k e p t q u i t e s t a t i o n a r y d u r i n g the /13/.

If lack of s t a t i o n a r i t y d u r i n g m e a s u r e m e n t s were t h e

experiments

m a j o r u n c e r t a i n t y of

t h e AR model, t h e n I would e x p e c t t h e m e a s u r e m e n t s to e x h i b i t g r e a t e r c o e f f i c i e n t s of v a r i a t i o n t h a n the c o m p u t e r

simulation, r a t h e r t h a n the o t h e r way r o u n d (especially

for series BH, for w h i c h it was m o s t difficult to keep basic conditions stable). However, a close look at t a b l e 2 a n d t a b l e 3 r e f u t e s t h e h y p o t h e s i s of lack of s t a t i o n a r i t y as a m a j o r source of t h e d i s c r e p a n c i e s b e t w e e n c o m p u t e r

simulation and experiment.

T h e s p e c t r a l densities in t h e wind t u n n e l were e s t i m a t e d w i t h e x t r e m e l y h i g h precision before f i t t i n g the AR models. If t h e s p e c t r a l d e n s i t y of t h e s t a t i o n a r y process to be s i m u l a t e d is a c c u r a t e l y known, the second i n c r e a s i n g t h e AR o r d e r / 1 3 / . d e n s i t y was v e r y close to

u n c e r t a i n t y can be r e d u c e d a r b i t r a r i l y by

In t h e case of series BT a n d BH, t h e AR s p e c t r a l

the tunnel spectral density.

U n d e r e a c h one of a n u m b e r of c o n d i t i o n s / 1 3 / ,

t h e t h i r d u n c e r t a i n t y is n o t v e r y

s i g n i f i c a n t w h e n i n t e g r a l scale e s t i m a t e s are studied.

535

Nevertheless, it will probably be possible to improve the agreement simulation and m e a s u r e m e n t further by doing something

between computer

about the two uncertainties

m e n t i o n e d u n d e r (2) and (3). /13/

Schrader,P., Die statistische Stabilitat gemessener i n t e r g r a l e r Langenmafle und

anderer

Windparameter,

(in

Ingenieurbau, R u h r - U n i v e r s i t a t

German),

Ph.D.

thesis,

Bochum, Bochum, Germany

Institut (1992),

fur

Konstruktiven

(will be available

soon)

Response Analyses on Along - Wind and Across - Wind Vibrations of Tall Buildings in Time Domain Authors : H. TSUKAGOSHI et al.

H.CHOI:In this paper, the comparison for one case of side ratio and aspect is shown. Have you confirmed the v a r i a b i l i t y of the method, i.e., eqs.(1) other cases h a v i n g d i f f e r e n t geometrical

ratio

(6), for

conditions ?

Response : I did not confirm the variability of the method I used. I think there are some restrictions on applying eqs.

(1)

~ (6)

to the model. In particular, the power

s p e c t r u m of the lift force is applicable to a model with a square plan and H/D= 3 ~ 1 0 in aspect ratio, because it is formulated on the basis of the data of the wind tunnel tests.

A.KAREEM

:

According to eq.(1), the spectrum corresponds to the fundamental mode.

Did you use some extrapolation to get spectra for higher modes, or restrict your analysis to the fundamental m o d e ? If you limited analysis to the fundamental mode, then the response for acceleration must

be underestimated. I also think that if your time domain

analysis is a lower model approach, then your simulation reduced to a o n e - degree of

freedom system in x and y directions.

Response:I

don't think the spectrum

(eq.(1))

corresponds to the fundamental mode.

It expresses the power s p e c t r u m of the f l u c t u a t i n g d r a g force at a h e i g h t z. So, I made response analyses by means of a model analysis for which each mode, up to the lOth mode, was taken into consideration so as not to underestimate the responses.

536

N.HOSOYA : I s

t h e r e a n y specific reason to h a v e chosen U = 6 0 m / s

(seems to be a little h i g h for a c c e l e r a t i o n a n a l y s i s ) ? U

60m/s

as tile top value

m a y be

suited [or

s t r u c t u r a l f r a m e d e s i g n loads.

Response : T h e

m e a n wind speed at a h e i g h t of 10m was set at 3 0 m / s e c . T h i s v a l u e

c o r r e s p o n d s to t h e basic d e s i g n wind speed used over a period of 50 y e a r s a c c o r d i n g to J a p a n e s e r e c o m m e n d a t i o n s . For t h e vertical profile for t h e m e a n wind speed, I used the power law

(with power law index a

0.25). As a consequence of these assumptions,

t h e m e a n w i n d speed at t h e top of t h e

building

(at a h e i g h t of 160m)

was set

at 6 0 m / s e c .

H.lOlWAl:What

m e t h o d did you use to s i m u l a t e t h e f l u c t u a t i n g d r a g ?

Response:The

t i m e h i s t o r i e s of t h e f l u c t u a t i n g d r a g a n d lift forces were

t h r o u g h t h e a p p l i c a t i o n of t h e a u t o - r e g r e s s i v e

simulated

m e t h o d for t h e f l u c t u a t i o n of the

w i n d s p e e d d e v i s e d b y I w a t a n i ( J o u r n a l of W i n d E n g i n e e r i n g , JAWE, No.11, pp.5 17 (in J a p a n e s e ) ) . Please r e f e r to his

paper.

Proposed Formulae for The Power Spectral Densities of Fluctuating Lift and Torque on Rectangular 3 - D Cylinders Authors : H.CHOI et aL

A.G,DAVENPORT:Isyumov

h a s i n v e s t i g a t e d t h e t o r s i o n a l forces on a v a r i e t y

of

b u i l d i n g shapes, etc. Have a n y c o m p a r i s o n s b e e n m a d e w i t h his w o r k ?

Response:The

case i n v e s t i g a t e d b y I s y u m o v et al.

(4)

was not t h e t o r s i o n a l force

b u t t h e l a r g e s t RMS t o r s i o n a l r e s p o n s e of a b u i l d i n g h a v i n g v a r i o u s sectional shapes, b a s e d on a e r o e l a s t i c model e x p e r i m e n t r e s u l t s or t h e p r o p o s e d f o r m u l a e . Because of lack of i n f o r m a t i o n , t h e s p e c t r a l c h a r a c t e r i s t i c s i n v e s t i g a t e d b y I s y u m o v et al. were not of a m o d e - g e n e r a l i z e d each other.

force. So it is not possible to c o m p a r e t h e two cases w i t h

537

Y.TAMURA :You mentioned in your conclusion that the proposed formulae are valid for aspect ratios 4 to 7. However, I cannot find the effects of the aspect ratio in your formulae. How did you check the effects of the aspect ratio ?

Response:It is generally true that the variation in spectral characteristics of fluctuating lift on rectangular cylinders are not serious for aspect ratio from 4 to 6, as indicated by Saunders et al.(6). Therefore, the formulae presented here can be said to be valid for aspect ratios from 4 to 6. Moreover, for aspect ratio 7, the validity of the formulae can be confirmed by the comparison results shown in Fig 3. (6)

Saunders,J.W., Melbourne,W.H., "Tall rectangular building response

excitation", Proc. 4th ICWE, Cambridge University Press,

to c r o s s - w i n d

1975, pp.369~379.

Numerical Simulation of Pressure Distributions Underneath Roofing Paver Systems Authors : Y.SUN et aL J.D.HOLMES : Your method is basically a steady - state solution. The problem similar to that of computing the response of interconnecting tubes measurement, in which fluctuating pressures are considered

is

for pressure

(the flow is assumed to

be compressible). Have you considered extending your results for fluctuating pressures ?

Response : Based on the assumption of flows between and beneath pavers, the numerical model is also expected to work for fluctuating pressures. However, this has not been validated.