12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

PARTICLE RESOLVED DNS STUDY OF TURBULENCE EFFECTS ON
HYPORHEIC MIXING IN RANDOMLY PACKED SEDIMENT BEDS

Shashank K. Karra
School of Mechanical, Industrial,and Manufacturing Engineering
Oregon State University
Corvallis, OR 97331
karras@oregonstate.edu

Sourabh V. Apte
School of Mechanical, Industrial,and Manufacturing Engineering
Oregon State University
Corvallis, OR 97331
sourabh.apte@oregonstate.edu

Xiaoliang He
Pacific Northwest National Laboratory
Richland, WA 99354
xiaoliang.he@pnnl.gov

Timothy D. Scheibe
Pacific Northwest National Laboratory
Richland, WA 99354
tim.scheibe@pnnl.gov

ABSTRACTtransport mechanisms across the SWI. Based on the characterization by Voermans et al. (2017, 2018); Grant et al. (2018)
turbulence is dominant near the SWI for ReK > 1 (figure 1a).
This study aims at using direct numerical simulations(DNS) to quantify the influence of varying ReK on turbulent
flow statistics near the SWI. Most of the prior experimental
and numerical studies have been done at higher Rek numbers(10 < ReK < 100) with uniformly distributed sediment beds.
Therefore this study aims at simulating turbulent flow over realistic randomly packed sediment bed arrangements with ReK
values (1 < ReK < 10) found in typical aquatic beds. This
study also aims at providing quantifiable data to improve theaccuracy of reach-scale modeling, which use the mean pressure and turbulent kinetic energy distributions at the SWI, obtained from a RANS based surface flow simulation, as boundary conditions at the top surface of a ground-water flow model
domain in a decoupled, sequential approach. In a previous
study by the authors (Karra et al., 2022), it was found that
turbulent and pressure fluctuations are better captured by considering at least the top layer of sediment bed at lower ReK
numbers. In the current work this is found to be true even at
higher ReK numbers with turbulent fluctuations due to bed heterogeneity peaking in the top layer for both lower and higher
ReK values. Importantly, the pdfs of pressure fluctuations forboth cases are found to be statistically similar, and well approximated by a fitted model, which can be used as a pressure
boundary condition at the SWI in reach scale models.
The rest of the paper starts with description of the simulation setup, parameters and mathematical formulation used in
the present work. A detailed description of the results and data
analysis including validation against experimental data is thenprovided followed by summarizing the major findings of thepresent work.

Pore-resolved direct numerical simulations (DNS) areused to investigate the interactions between stream-water flow
turbulence and groundwater flow through a porous sediment
bed in the hyporheic zone. Two permeability Reynolds numbers (2.56 and 5.17), representative of aquatic systems andrepresenting ratio of permeability to viscous length scales,
were simulated to understand its influence on the momentum
exchange at the sediment-water interface (SWI). A doubleaveraging methodology is used to compute the Reynolds
stresses, form-induced stresses, and pressure fluctuations. It
is observed that both shear layer and turbulent shear stresspenetration increases with ReK . Reynolds and form-induced
bed-normal stresses increase with ReK . The peak values of theform-induced stresses for the lower (2.56) and higher (5.17)
ReK happen within the top layer of the sediment bed. The sumof turbulent and form-induced pressure fluctuations, analyzed
at their respective zero-displacement planes, are statisticallysimilar and can be well approximated by a t location-scale
distribution fit providing with a model that could potentiallybe used to impose boundary conditions at the SWI in reachscale simulations.

INTRODUCTION
The interchange of mass and momentum between surface
water and ground water occurs in the porous bed beneath thestreams, termed as the hyporheic zone. The exchange of massand momentum (hyporheic exchange) of solutes such as chemicals and pollutants, dissolved oxygen, nutrients, and heat
across this zone is a critical component in sustaining diverse
aquatic ecosystems. The penetration of mean and turbulent
flow within the porous bed and near bed pressure fluctuations
have been identified (Hester et al. (2017)) to play a crucial inunderstanding this hyporheic exchange. While there are many
parameters which characterize exchange of mass and momentum across the sediment water interface (SWI), the permeability Reynolds number, representing the√ ratio between the permeability scale to the viscous scale K/(ν/uτ ), is typicallyused to identify different flow regimes based on the dominant

SIMULATION DOMAIN AND PARAMETERS
The computational domain for the permeable sediment
bed used in this study consists of a doubly periodic domain
with four layers of randomly packed, mono-dispersed sediment grains at the bottom to capture the turbulence penetration1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1. Parameters used in present pore-resolved direct numerical simulations where, D p is the sphere diameter, δ isthe free surface height, Hs is the sediment depth, and θ is theporosity.

VV

PBL

PBH

ReK

2.56

2.56

5.17

Reτ

180

270

540

θ

0.41

0.41

0.41

Hs /δ

1.71

1.14

1.14

D p /δ

0.43

0.29

0.29

(Lx , Lz )/δ

(4π,2π)(4π,2π)(2π,π)∆x+ = ∆z+

3.01

3.01

4.06

∆y+

0.95

0.95

0.95

(a)

(b)

Figure 1. (a) Effective dispersion coefficient versus
Rek (based on Voermans et al. (2017); Grant et al. (2018)),(b) schematic of permeable bed showing height of roughness
element layers, Hs , and boundary layer thickness, δ .

and unsteady, inertial flow (figure 2, Karra et al. (2020); Karra& Apte (2020)) . The random packing of mono-dispersed,
spherical particles within a doubly periodic box is generatedusing the code developed by Dye et al. (2013).
Turbulent flow over a permeable bed can be characterized by the permeability Reynolds number (ReK ), the turbulent
Reynolds number (Reτ ), the ratio of sediment depth to the freesurface height (Hs /δ ), the ratio of the sediment grain diameterto the free-surface height (D p /δ ), bed porosity (θ ), and the domain lengths in the axial and spanwise directions normalized
by the free-surface height (Lx /δ , Lz /δ ) (figure 1b). Table 1
shows detailed simulation parameters for the cases used to investigate the structure and dynamics of turbulence over porous
sediment bed. Three permeable bed cases are simulated; caseVV, case PBL, and case PBH. Case VV is used to verify andvalidate the DNS simulations of turbulent boundary layer flow
over a sediment bed with experimental data from Voermans
et al. (2017). Permeable bed case with porosity (θ ) = 0.41,
ReK = 2.56 and Reτ ∼ 180 match those of Case L12 in Voermans et al. (2017), and is within range for realistic aquatic
sediments beds. Case PBL is simulated at lower ReK value of
2.56 and case PBH is simulated at a higher ReK value of 5.17.
The grid resolutions required for these configurations are
based on two main considerations: (i) minimum bed-normal
grid resolution near the bed, and (ii) minimum resolution required to capture flow over spherical particles. For DNS ofboundary layers, the bed-normal grid resolution in wall units
should be ∆y+ < 1, in order to accurately capture the bed shear
stress in the turbulent flow. The grid resolutions in the axial
and spanwise directions are typically 3–4 times coarser, following the smooth channel flow simulations by Moser et al.
(1999). Note that, the roughness features and permeability are
known to break the elongated flow structures along the axial
direction in smooth walls, reducing the inhomogeneity in thenear-bed region. This flow feature is thus anticipated to alleviate the bed-normal grid resolution requirement near the bed.
To capture the inertial flow features within the pore and aroundFigure 2. Permeable bed with four layers of sediment particles (inset shows close-up view in xy-plane)

spherical particles, grid refinement studies conducted on flow
over single sphere (not shown) are used. Accordingly, roughly
180 grid points for the PBL case and 308 points for the PBH
case are used in the bed-normal direction within each sedimentparticle in the top two layers. In the x and z directions, uniform grid with 26 and 38 grid points are used to resolve thegrain geometry for the PBL and PBH cases, respectively.

Numerical scheme
The numerical approach is based on a fictitious domain
method to handle arbitrary shaped immersed objects without
requiring the need for body-fitted grids (Apte et al., 2009).
Cartesian grids are used in the entire simulation domain, including both fluid and solid phases. An additional body force
is imposed on the solid part to enforce the rigidity constraint
and satisfy the no-slip boundary condition. The absence ofhighly skewed unstructured mesh at the bead surface has beenshown to accelerate the convergence and lower the uncertainty (Finn & Apte, 2013). The following governing equations are solved over the entire domain, including the region
within the solid bed, and a rigidity constraint force, f, is applied that is non-zero only in the solid region. The governing2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

(d)

(e)

Figure 3. Comparison of (a) mean velocity and (b) streamwise, and (c) wall-normal components of Reynolds stress tensor, (d) streamwise and (e)wall-normal components of form-induced stress tensor. Experimental data by Voermans et al. (2017) ( ), emulating experimental sampling ( ), DNS ( ).

double averaging procedure.
Pore-resolved direct numerical simulations of turbulent
boundary layer flow over a sediment bed were first validatedwith experimental data from Voermans et al. (2017). Permeable bed case with porosity of 0.41, ReK = 2.56 and Reτ ∼ 180
matches with Case L12 in Voermans et al. (2017),which iswithin the range for realistic aquatic sediments beds. For thisvalidation study, the simulation domain used is same as in the
DNS study of Shen et al. (2020).
The double-averaged (DA) mean velocity profile normalized by free-surface velocity Uδ is shown in figure 3a. Excellent agreement is seen between the DNS data and experimental measurements. Figures 3b, and 3c, show a comparison of DA turbulence intensities, namely stream-wise, andwall normal stresses. Again very good agreement between
DNS and experiment is observed. The form-induced intensities or dispersive stresses normalized by uτ are shown infigures 3d, and 3e. Upon initial observation, noticeable differences are observed between current DNS and experimental
results for dispersive stresses which can be explained as follows. Firstly, as mentioned in the previous section, spatial
averaging is carried out over an entire x − z plane at a given
y location for DNS results. While for the experimental data,
spatial averaging was performed over three different spanwise
locations over six different measurements. To quantify the differences in the sampling procedures between the experiments
and DNS, the experimental sampling process is replicated in
the DNS data whereby spatial averaging is carried out at a few
finite uncorrelated spanwise locations and repeated over different streamwise locations. A family of curves, shown by greysquares, indicates the associated uncertainty in the processingof the experimental data. The averaged experimental and DNS
data are within this scatter for all axial locations. Secondly,
it is has been reported in literature Nikora et al. (2002); Fanget al. (2018) that the spanwise averaging is highly sensitive tothe geometry at the sediment-water interface. For the present
DNS, only the mean porosity of the randomly distributed arrangement of mono-dispersed spherical particles is matchedwith the experimental geometry. However, the exact sedimentgrain distribution in the experiments is unknown and is likely

equations are given as:ρf∇ · u = 0,∂u
+ (u · ∇) u = −∇p + µ f ∇2 u + f
∂t

(1)
(2)

where u is the velocity vector (with components given byu = (ux , uy , uz ), ρ f the fluid density, µ f the fluid dynamic viscosity, and p the pressure. A fully parallel, structured, collocated grid solver has been developed and thoroughly verified
and validated for a range of test cases including flow over acylinder and sphere for different Reynolds numbers, flow overtouching spheres at different orientations, flow developed byan oscillating cylinder, among others. The details of the algorithm as well as very detailed verification and validation studies have been published (Apte et al., 2009; Finn & Apte, 2013).

RESULTSIn this section validation of DNS simulations (VV case)is shown first followed by results for different ReK cases, PBL
and PBH.

Comparison with experimental data
Since the flow properties are highly spatially heterogeneous near rough sediments, double averaging procedure (Raupach & Thom, 1981) is used, wherein spatial averaging is performed along with time averaging,
φ (x,t) = ⟨φ ⟩(y) + φe(x) + φ ′ (x,t)(3)

where φ is an instantaneous flow variable, ⟨φ ⟩R is the intrinsic
spatial average in the (x, z) plane, ⟨φ ⟩ = 1/A f A f φ dA (whereA f is the area occupied by the fluid), φ is the temporal average, φ ′ = φ − φ is the instantaneous turbulent fluctuation and
φe = φ − ⟨φ ⟩ is the form-induced or dispersive fluctuation. Accordingly, Reynolds stresses, form-induced stresses and , pressure fluctuations shown in this section are computed using the3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 2. The von-Kármán constant (κ), zero-displacement
thickness (d), and equivalent roughness height y0 normalizedby ν/uτ . Brinkman-layer thickness (δb ) and Reynolds-shear
penetration depth are normalized by the free surface height δ .

Figure 4. Schematic showing positions of sediment crest(y = 0), zero-displacement plane (y = −d), and particle diameter (D p ).

κ

d+

y+
0

δb /δ

δ p /δ

PBL

0.32

47.0

6.65

0.197

0.382

PBH

0.31

96.0

8.87

0.248

0.44

Table 3. Peak value of normalized Reynolds and forminduced stresses for the PBL and PBH cases. Axial and bednormal Reynolds stresses as well as the form-induced bednormal, and shear stresses are normalized by u2τ , whereas theform-induced pressure fluctuations are normalized by ρu2τ .

different compared to that used in DNS. This difference, especially near the top of the bed can also contribute to differencesin the form-induced or dispersive stresses.

Zero-displacement plane and shear penetration
In turbulent flows over rough walls and permeable beds
the log-law has the following form


U(y)
1
y+d
= log
uτ
κ
y0

Case

(4)

where κ is the von-Kármán constant, d is distance between
the zero-displacement plane and the top of the sediment crest,
and y0 is the equivalent roughness height. Nikora et al. (2002)
define d as the level that large-scale turbulent eddies feel as
the bed origin and thus, their sizes linearly scale with distancefrom this level. The equivalent roughness height, y0 , is related to a measure of the size of the roughness elements. A
schematic of the domain with the location of the sediment crestand the zero-displacement plane is shown in figure 4 wherey = 0 is chosen at the sediment crest in the domain. For comparison of flow statistics between the PBL and PBH cases thevirtual origin is chosen at the zero-displacement plane, y = −d.
The fitting procedure (shown in Karra et al. (2022)) describedby Breugem et al. (2006) is used to obtain the values of d andy0 and are shown in table 2.
The depth of the shear layer penetration, known as the
Brinkman layer thickness, δb , is defined as the vertical distance from the zero-displacement plane (y = −d) to the y location, inside the bed, which separates the constant or Darcy
velocity region from the shear layer above it. Following theprocedure by Voermans et al. (2017), δb is calculated as they location (measured from y = −d) at which the difference
between the local mean velocity and Darcy velocity, U p , has
decayed to 1% of the velocity value at the zero displacement
plane, Ui , i.e ⟨u⟩(y)y+d=−δb − U p = 0.01(Ui − U p ). Table 2
shows the δb values for both the PBL and PBH cases. Shearlayer penetration increases ReK and is of the same order as thatof the sediment grain diameter for both cases.
The penetration depth of the turbulent shear stress into
the bed, δ p , defined as , ⟨u′ v′ ⟩y=−δ p = 0.01⟨u′ v′ ⟩i , is shown intable 2. The value of δ p increases with ReK , even though theporosity of the beds for PBL and PBH cases remains the same.

Case⟨u′2 ⟩+⟨v′2 ⟩+

⟨e
v2 ⟩+

⟨e
uve⟩+

⟨ pe2 ⟩+

PBL

3.51

1.02

0.25

-0.21

3.69

PBH

3.09

1.17

0.48

-0.3

16.94

(figure 5a ), is greater for the PBL case than the PBH case. Thisis because at lower permeability Reynolds numbers, ReK , theturbulent fluctuations into the wall (i.e., sweeps) are redirected
into wall parallel components due to lower bed permeability.
The bed-normal velocity fluctuations, ⟨v′2 ⟩+ , (figure 5b ) for
the PBH case are greater than the PBL case, and therefore increase with ReK , suggesting both an increase in magnitude andpenetration of flow turbulence into the sediment bed. The peakvalues for the Reynolds stresses, ⟨u′2 ⟩+ and ⟨v′2 ⟩+ , happen
very close to the sediment crest level and are shown in table 3.
Compared to the PBH case the peak value of ⟨u′2 ⟩+ and ⟨v′2 ⟩+for the PBL case is 11.9% greater and 12.8% lower.
Figures 5c,d show the bed-normal and shear form induced
stresses and figure 5e shows the form-induced pressure fluctuations. Compared to the Reynolds stresses the peak valueof form-induced stresses occurs significantly below the sediment crest. Similar to the Reynolds stress, the bed-normal
form-induced stress, ⟨ev2 ⟩+ , increases with ReK , however thepercentage increase is larger and the value of ⟨ev2 ⟩+ for PBH
case is 47.9% greater than the PBL case, indicating that spatial heterogeneity at the roughness element length scale enhances both the turbulence intensity and penetration. The
form-induced shear, ⟨euve⟩+ , also increases with ReK , with thepeak value for PBH case being 30% greater than the PBL case.
In case PBH the form-induced stress has secondary peak in thefree-stream region. These peaks in form-induced stresses havebeen reported by Fang et al. (2018) and are thought to be induced by time-averaged structure of secondary currents. Coleman et al. (2007) reported measuring these secondary currents
in the free stream. They added these secondary current terms(namely, the integral from sediment crest to the free surface of,e z)ρ⟨w⟩∂ ⟨u⟩/∂ z + ρ⟨v⟩∂ ⟨u⟩/∂ y + ρ∂ ⟨u′ w′ ⟩/∂ z + ρ∂ ⟨euw⟩/∂
to the total stress balance equation and highlighted their influence on momentum transfer. The peak value for form-inducedpressure fluctuations, ⟨ pe2 ⟩+ , in the PBH case is 78.2% larger
than the peak in PBL case. This is an important observation
as it points to high sensitivity of spatial heterogeneity induced

Turbulence structure
Figures 5a,b show the streamwise and bed-normal
Reynolds stresses for the PBL and PBH cases. The doubleaveraged variables are normalized by the friction velocity, u2τ
(pressure by ρu2τ ), and y is shifted by d and then normalized by
δ , effectively making virtual origin the same for both the cases.
The axial component of turbulent velocity fluctuation, ⟨u′2 ⟩+4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

(d)

(e)

Figure 5. Comparison of Reynolds and form induced stress profiles for PBL ( ) and PBH cases ( ). (a) streamwise, and (b)wall-normal components of Reynolds stress tensor, (c) wall-normal and (d) shear stress components of form-induced stress tensor and(e) form-induced pressure fluctuations. Horizontal lines show the crest of sediment bed for PBL ( ), and PBH ( ) cases.
Table 4. Higher order statistics for turbulent and form induced pressure fluctuations, showing the mean µ(·), standard
deviation σ (·), skewness Sk(·), and kurtosis Ku(·).

pressure fluctuations on ReK . This increase in magnitude ofpressure fluctuations at higher ReK can result in decrease inparticles/contaminants residence time inside the sediment bed.
Most importantly, the form-induced stresses and pressure fluctuations peak within the top layer of the sediment bed for bothlow (2.56) and high (5.17) ReK numbers. This further extends the observations reported by the authors in their priorwork (Karra et al., 2022), on the need to include atleast the
top layer of the sediment bed in reach scale RANS models tobetter capture the influence of bed roughness on momentum
exchange near the SWI to, higher ReK numbers.

Turbulent pressure fluctuations
Pressure fluctuations at SWI play a critical role in hyporheic transport even for a flat bed. Specifically, pressure
fluctuations due to turbulence are conjectured to have significant impact on mass transport within the hyporheic zone as
it can directly influence the residence times through turbulent advection. Pressure fluctuation statistics for the PBL and
PBH cases are compared at their respective zero-displacement
planes, y = −d.
Figure 6a shows the pdfs of the sum of the turbulent, p′ ,and form-induced, pe, pressure fluctuations normalized by theirrespective standard deviations for the PBL and PBH cases.
The pdfs for both cases are statistically similar, closely approximating a normal distribution with heavier tails. This is a very
important observation as this means that the pressure behavior inside the bed can be approximated with a model acrossa range of permeability Reynolds numbers, ReK , which are
typical for aquatic sediment beds. The pdf distributions in figure 6a was fitted with a t location-scale distribution, typicallyused for modeling data distributions with heavier tails. Thismodel can be used as a boundary condition at the SWI in reachscale models. The parameters that define the t location-scale
distribution model are the mean, µ, shape factor, ν, which canbe understood as an estimator for skewness and kurtosis, andσ 2ν.
variance, defined using the standard deviation, σ and ν as ν−2
Figures 6b,c show the pdf distributions for the turbulent,p′ , and form-induced, pe, pressure fluctuations separately forboth PBL and PBH cases. It can be observed that the pdf for

PBL

PBH

µ( pe + p′ ), σ ( pe + p′ )-0.03, 2.77

-0.16, 13.0

Sk( pe + p′ ), Ku( pe + p′ )0.75, 6.87

0.77, 6.67µ(p′ ), σ (p′ )-0.13, 2.38

0.09, 10.64

Sk(p′ ), Ku(p′ )-0.15, 5.25

-0.11, 7.27µ( pe), σ ( pe)-0.03, 1.60

-0.16,7.28

Sk( pe), Ku( pe)1.70, 8.08

2.13, 10.21

turbulent fluctuations is closer to normal distribution behavior
whereas the form-induced pressure fluctuations have skewed
distributions showing how the bed surface variations influence
local pressure distribution. Table 4 confirms that skewness,
and kurtosis for pe is higher than p′ .

CONCLUSIONS
Pore-resolved direct numerical simulations of turbulent
boundary layer over a porous sediment bed were performed at
two permeability Reynolds numbers (2.56 and 5.17). The key
findings from these numerical experiments are summarized:(i) The shear layer (Brinkman layer) thickness and turbulent shear stress penetration increase with ReK and are on theorder of the sediment grain diameter.
(ii) The Reynolds and form-induced stresses are influenced by the permeability Reynolds number, ReK . The streamwise Reynolds stress peak value decreases with increase in
ReK whereas, the wall-normal Reynolds stress increases. Both
bed-normal and shear form-induced stresses increase with
ReK . While the peak values for the Reynolds stresses happen

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

(b)

(c)

Figure 6. PDFs of (a) sum of turbulent p′ , and form-induced pressure fluctuations, pe (the symbols ( ) show the ”t location-scale
distribution” fit), (b) p′ fluctuations and (c) pe fluctuations. Legend: PBL ( ) and PBH ( ).
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near the sediment crest, the form-induced stress peak values
are observed inside the sediment bed. Importantly, the peakvalues of form-induced stresses, for both PBL and PBH cases,
occur within the top layer of the sediment bed, highlighting
the importance of including at least the top layer of the sediment bed in reach scale models to better capture the influenceof bed roughness on momentum exchange near the SWI evenat higher ReK values.
(iii) The sum of turbulent and form-induced pressure
fluctuations, analyzed at their respective zero-displacement
planes, are statistically similar and can be well approximatedby a t location-scale distribution fit. This is of critical importance as it can be potentially used as a boundary condition at
the SWI in reach scale RANS simulations.

ACKNOWLEDGEMENTS
SVA and SKK gratefully acknowledge funding from US
Department of Energy under award number DE-SC0021626
as well as US National Science Foundation award #205324.
Frontera computing resources used under NSF’s LeadershipResources Allocation (LRAC). XH and TDS acknowledge
funding from the DOE Office of Biological and Environmental Research, Environmental System Science program,
through the PNNL River Corridor Scientific Focus Area
project (https://www.pnnl.gov/projects/river-corridor).

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