Erosion of biofilm-bound fluvial sediments

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NGEO1891 Erosion of biofilm-bound fluvial sediments Elisa Vignaga, David M. Sloan, Xiaoyu Luo, Heather Haynes, Vernon R. Phoenix and William T. Sloan Mathematical Model In our experiments we observe that the biofilm and sediment form a composite elastic material. When we increase the flow rate in our hydraulic flume we observe oscillations in the membrane which grow and ultimately cause the membrane to rip. The maximum wavelength of the oscillation is L, the length of the membrane. Our motivation for modelling is to identify the modes of oscillation that are most unstable at a wide range of realistic flow rates (characterized by the non-dimensional Reynolds number) and hence determine whether the failure mode that we observe could potentially be generic for biofilm bound sediments. Schematics of two dimensional slices through our experimental flumes are shown in Figures 1A and 1B of the main text. Figure 1A is for a flume where a glass plate has been lowered on to the surface and hence the flow is approximately described by Poiseuille flow. Figure 1B represents open channel flow which is typical of real fluvial systems; we assume that the upper boundary experiences atmospheric pressure and no applied shear stress. We assume that the flow can be described by the Navier-Stokes equations and the motion of the biofilm-sediment composite is described by equations of motion NATURE GEOSCIENCE 1

2 for an elastic membrane that is loosely attached to the substratum by elastic filaments of biofilm. Flow equations are developed below for both flumes. In addition, for the purpose of comparison, we also develop the equations for flumes where the bed is fixed. 1 Flow Equations 1.1 Poiseuille flow in a capped flume. We consider flow with streamwise direction confined between planes located at. It is convenient to use non-dimensional variables,,,, and, where - denotes the dimensional variables. Here, h is half the depth of the channel, U 0 is some characteristic speed, is the pressure, ρ is the constant fluid density, and and denote streamwise and normal velocity components. In terms of the non-dimensional variables, the Navier-Stokes equations take the form 0, (1) 1, (2) 1, (3) where the non-dimensional parameter is the Reynolds number and is the kinematic viscosity coefficient. For plane Poiseuille flow with rigid boundaries at and Reynolds number defined with U 0 equal to the centreline velocity, the non-dimensional streamwise velocity has the form () 1. (4) In this case the flow is driven by a constant pressure gradient.

3 In the compliant wall problem considered here we investigate the stability of the flow by considering the growth of perturbations about Poiseuille flow; this effectively assumes that the pressure gradient has no effect on the compliant wall. Let (, ) (() (,, ),(,, )) and 2/ (,, ), where, and represent small disturbances in the velocity components and pressure. We now substitute into equations (1)-(3) and ignore quadratic terms in the small disturbances to obtain 0, (5) 1, (6) 1. (7) We now introduce a stream function (,, ) to satisfy equation (5) and investigate stability by considering wave-like disturbances of the form (,, ) () (), (8) () (), (9) where is the wave number and c is the wave speed. The velocity components are now given by (), (), (10) and if, and are replaced by the wave-forms in (8)and (9) we obtain ( ) 1 ( ), (11) () ( ). (12)

4 Here, the dash denotes differentiation with respect to y. The variable is readily eliminated to give the Orr-Sommerfeld equation ( )( ) 1 ( 2 ). (13) When equation (13) is coupled with appropriate homogeneous boundary conditions, solutions exist when c is an eigenvalue of the system. If the imaginary part of c is positive then the corresponding wave-form grows with time and the solution is linearly unstable Capped Flume Top Boundary During our first set of experiments, in order to view the bed with a high-speed camera without interference from perturbations of a free surface, a glass plate was placed at the water surface. Therefore, at the top boundary in our capped flume we have no-slip conditions. It is readily seen from equation (10) that, in terms of, the no-slip conditions at the top boundary are (1) 0, (1) 0. (14) Bottom boundary: Case 1 Fixed Lower Boundary In our analysis we compare the results for a flexible membrane on the bed of the flumes with those for a fixed, inflexible boundary. So consider, first, the simpler condition where the bed is fixed and inflexible. In this case, at y = -1 we have no-slip, so we impose (1) 0, (1) 0. (15) Bottom boundary: Case 2 Flexible Elastic Membrane Membrane motion defines the behavior of the lower boundary in our flow model and the feedback between the flow and the membrane motion has an effect on the stability of small perturbations in the flow.

5 The effect of compliant walls on flow instabilities for a spring-backed plate that is constrained to move only in the vertical direction of a membrane has been given in Carpenter and Garrad (1986) 1. Using their approach, we may write the equation of motion of the membrane at the lower wall as, (16) where (, ) is the vertical displacement of the lower wall from its mean position y h, is the mass of the membrane per unit area, is the membrane damping coefficient, is the flexural rigidity, is the spring stiffness, is the longitudinal tension in the membrane per unit width, is the perturbation fluid pressure at the mean wall location y. As in Carpenter and Garrad (1986) 1, we assume that the displacements are small. Again, we use non-dimensional variables x and t together with, and the non-dimensional parameters,,, and. Equation (16) may now be written in non-dimensional form as 1 (17) To cast this equation in a form that may be utilised as a boundary condition in the eigenvalue computation we use the wave-form for as in (9) and assume that may be represented similarly as (, ) () (18) This enables us to write (17) as 1 ( ) (1). (19) To first order in the small disturbances, the tangential and normal components of the fluid velocity at the point (, 1, ) on the membrane

6 are, respectively, (1 ) (,1, ) and (, 1, ). The corresponding velocity components of the membrane at this point are zero and. Now match the corresponding components, utilise the wave-forms (10) and (18) and linearise about y = -1 to obtain (1) (1) 0, (1). (20) If is eliminated from equations (20) we obtain the membrane boundary condition (1) (1)(1) 0. (21) For the Poiseuille flow profile () 1 and hence (1) 2(1) 0. (22) To obtain a second boundary condition at the membrane, set y equal to -1 in (11) and utilise (20) and (21) to obtain the condition (1) 1 (1) (1). (23) If this is substituted into (19) we obtain, using (20), the boundary condition 1 ( )(1) (1) (1). (24) Note that the quadratic term in c appearing in the boundary condition (24) may be removed. It is notationally simpler to write (24) as (1) (1) (1) (1) (1). (25) where ( ),,, and. If (1) in(25) is replaced by (1) we see that (25) may be replaced by

7 (1) 2 (1) (1) (1), (26) which is linear in the eigenparameter c. The task is now to solve (13) subject to boundary conditions (14), (22) and (26) Summary of Capped Flume Model Unstable modes are described by the solution of equation (13) subject to boundary conditions (14), (22) and (26). The numerical solution scheme is given below (Section 2). In the computations, the dimensional parameters R, T m, d, B and K are specified (Table S2), and the eigenvalues, c, are obtained for a range of values of. The flow associated with the parameter set is deemed to be unstable if, for any value of, an eigenvalue c is found with positive imaginary part. The results are displayed in the main text (Figure 1C) as curves on the (, ) plane on which the flow is marginally stable: that is, at any point on such a curve the imaginary part of the least stable eigenvalue is zero Justification of Membrane Parameters The biofilm is anisotropic and is stiffer in the horizontal plane than in the vertical; when the biofilm is picked up it comes away in lateral sheets rather than clumps. We estimated that there are approximately 10 times more bacterial filaments lying horizontally than vertically and hence the vertical modulus of elasticity would be approximately th of that measured in our load test. Thus the spring stiffness is approximated by, where l f is the length of the filaments that extend downward in the y direction and E is the Young s modulus measured in our load test. The flexural rigidity is proportional to the product of E and the cube of the membrane thickness,, such that ( ), where is Poisson s ratio. All our membranes were approximately 0.002m thick. Despite the fact that for our biofilm composites

8 E is relatively low and the membrane is thin, the flexural rigidity plays an important role in the membrane behavior. This is a finding that is mirrored by other studies. So, for example, in modeling mitral valves (with thickness as little as 0.125mm) Luo at al. (2012) 2 found that the experimental and modeling results could only be matched by including flexural rigidity. We have little information on the pretension in our elastic-composite materials but have assumed that there will at least be some tension caused by movement of sediment grains once the bacterial filaments in the biofilm have attached. Therefore, non-dimensional was set to an arbitrary small value of /10. As the membrane flexes and is consequently stretched the stress will increase over and above any pre-tension, T, resulting in extensional stiffening of the membrane. The small deflection model that we use does not account for these secondary nonlinear affects, which may act to stabilize the motion of the real membrane. We use the damping coefficient,, to demonstrate the effects on stability as energy is dissipated by the membrane motion. The density of the both the sand-biofilm composite and the glass bead-biofilm were measured at 1700 kg/m 3. Hence the mass per unit area,, was kg/m Open Channel Flow Equations. The development of the model for the open channel flume (Figure 1B) is similar to that for the capped flume. However, the laminar flow profile in an open channel, upon which we superimpose oscillations, differs from Poiseuille flow and, therefore, we need to determine the form of this profile. In addition our boundary conditions differ Open channel flow profile. To determine the laminar flow profile down an inclined plane (Figure 1.B) we use the Navier-Stokes equations in the non-dimensional variables (1) - (3) with force terms sin and cos added to the right hand side of

9 (2) and (3), respectively. The g is the gravitational constant and is the slope of the flume. We assume (, ) ((),0) and (), and impose the boundary conditions 0 at y = -1 (no slip) 0 at 1 (no applied shear stress) dimensionless) at 1 (atmospheric pressure, to determine the laminar flow profile. Here is the dimensionless shear stress, which is readily shown to be. (27) It is a straightforward task to show that cos (1). (28) and 2 sin (4 ( 1) ). (29) If we scale u using 2sin then we get the laminar flow profile () (1), (30) scaled so that U(+1) = 1 and U(-1) = The growth of perturbations about the laminar open-channel profile We use a linear stability analysis, identical to that used for the capped flume, to assess the stability of perturbations about the laminar flow profile. Let (, ) (() (,, ),(,, )) and () (,, ), where, and represent small disturbances in the velocity components and pressure.

10 By substituting into equations (1)-(3) (with the gravitational terms sin and cos included) ignoring quadratic terms in the small disturbances and assuming wave forms for, and, we arrive at the Orr-Somerfield equation, repeated here for completeness, ( )( ) 1 ( 2 ). (31) Open Channel Top Boundary At the upper open-surface we impose zero shear stress and atmospheric pressure. Zero shear stress at the upper surface requires 0 at 1. Now we use (,) and note that (1) 0 to obtain the zero shear stress condition at y =1 in the form 0. (32) This yields, on inserting the wave forms (10), 0 at 1. (33) The next boundary condition comes from constant pressure at y = 1. Since (1) we now require (, 1, ) 0, and from the x-momentum equation (11) we derive an equation from the condition 0. We find ((1) ) (1) 1 (1) (1). (34) With U(1) = 1 this becomes (1) 1 (1) (1) (1). (35)

11 1.2.4 Open Channel Bottom Boundary: Case 1 Fixed and Inflexible. In this case, the condition is the same as in the capped flume; at y = -1 we have no-slip, so we impose, (1) 0, (1) 0. (36) For the fixed lower boundary we have to solve (31) subject to (33), (35) and (36) Open Channel Bottom Boundary: Case 1 Elastic Membrane. For the open channel the bottom boundary conditions are developed in exactly the same way as for the capped flume. However, the laminar flow profile has changed so the first condition (1) (1)(1) 0 (37) for the open channel flow profile, which is () 1 (1), becomes (1) (1) 0. (38) The second condition, which is derived from considering the pressure and momentum on the membrane remains 1 ( )(1) (1) (1). (39) Again we can remove the quadratic term in c using (38). This enables us to replace (39) by (1) ( ) (1) (1) (1), (40) where W, Z, P, S and Q are as previously defined.

12 1.2.6 Summary Open channel Unstable modes are described by the solution of equation (31). When there is a flexible bed the boundary conditions are (33), (35), (38) and (40). When the bed is fixed the boundary conditions are (33), (35) and (36). 2 Numerical Solution We illustrate the numerical solution by considering the capped flume with a flexible bottom boundary. This means solving equation (13) subject to boundary conditions (14), (22) and (26) To solve the eigenvalue problem we use a Chebyshev pseudospectral method related to that presented in Huang and Sloan (1994) 3. We use the Gauss- Lobatto collocation points ( 1), 1,2,, (41) 1 where N is a positive integer. We denote the Lagrange interpolation polynomials for this set of points by (), 1,2,,. (42) This set of polynomials of degree (N-1) satisfies the condition ( ), (43) where is the Kronecker delta. The function () is approximated by the polynomial () 1 1 () (44) where,,, are constants. Note that

13 () () ( 1) () 1 (45) and it follow from (43) that ( ) ( ) 0. Hence, the polynomial of degree N, (), satisfies the boundary condition (14) at 1. Note also that ( ) for k = 2,3,,N so is an approximation to ( ). Since ()contains (N-1) unknown constants we require (N-1) equations to construct the approximate solution. Two equations are given by the boundary conditions (22) and (26) and the remaining (N-3) are provided by approximating the differential equation (13) at a subset of the nodes,,,. The pseudospectral approximation to (13) at a typical node is given by the collocation equation 1 ( ) 2 ( ) ( )2 ( ) (1 ) ( ) ( ) ( ) ( ). (46) Here we have noted that () 1. The derivatives of () up to order 4 are required at the nodes,,,. From (44) we see that () () which gives ( ) 1( 1,1), (47) where 1( 1,1) denotes the above derivative evaluated at. Here we use 1(, ) to denote the element in row m and column n of an ( 1) ( 1) matrix D1, which is known as a first order pseudospectral differentiation matrix. Note that 1( 1,1) ( ) ( 1) ( ) 1

14 ( 1)1( 1,1) 1 (48) where 1( 1,1) is the element in row (k-1) and column (j-1) of a matrix M1. Hence, we may write 1 ( 1) (49) where I is the unit matrix and DL is a diagonal matrix with elements ( 1), ( 1),,( 1). The matrix M1 is obtained by deleting the first row and first column from and matrix () which is known as the firstorder Chebyshev pseudospectral differentiation () (, ) is the value of () at for j,k = 1,2,,N. matrices. The element By differentiating () up to order 4, and proceeding as above, it is readily shown that the differentiation matrices D2, D3 and D4 are given by 2 (2 1 2) 3 (3 2 3) 4 (4 3 4) where M2, M3, and M4 are obtained by deleting the first row and column from (), () and () respectively. (), () and () are the Chebyshev pseuadospectral differentiation matrices of order 2, 3 and 4. Many authors have presented efficient and accurate algorithms for constructing pseudospectral differentiation matrices. The eigenvalue problem considered here was coded in MATLAB, and the matrices () ( 1,2,3,4) were obtained using the MATLAB function chebdif which is presented in Weideman and Reddy (2000) 4.

15 Returning to the collocation equations we note from (47) that the vector containing the values ( ), 2,3,, is given by 1 where,,,, with analogous results for derivatives of orders 2,3,and 4. Also, the vector containing values of ( ), 2,3,, is given by or. The (N-1) collocation equations for the nodes,,, may now be written in matrix form as 1 (4 2 2 ) 2 (50) where 2 and Y is the diagonal matrix with elements (1 ), (1 ),,(1 ). This system has the form 1 1, (51) where A1 and B1 are ( 1) ( 1) matrices. We require (N-3) collocation equations, and to this end we drop the first and last rows of A1 and B1 and retain the collocation equations at the nodes,,,. To incorporate boundary conditions (22) and (26) we append two rows to the reduced A1 and B1 based on approximating (1) and (1) by 1( 1, ) and 3( 1, ), respectively, for 1,2,, 1. The final generalised linear eigenvalue problem has the form (52) where A and B are (1)(1) matrices. It should be noted that the condition numbers of A and B especially A may be huge in some cases. For such cases, the solution of (52) by a generalised eigenvalue routine will suffer from roundoff error. As in Huang and

16 Sloan (1994) 3 we apply a simple preconditioner to reduce the condition number of A prior to the computational solution. Equation (52) is premultiplied by a diagonal matrix L which has diagonal elements 1 (, ), 1,2,,1 where A(k,k) is the element in row and column k of A. The final system is. (53) The eigenvalues, c, of (53) are computed using the MATLAB function eig. 1 Carpenter, P. W. & Garrad, A. D. The hydrodynamic stability of flow over kramertype compliant surfaces.2. Flow-induced surface instabilities. J Fluid Mech 170, (1986). 2 Luo, X. Y. et al. Effect of bending rigidity in a dynamic model of a polyurethane prosthetic mitral valve. Biomechanics and Modeling in Mechanobiology 11, (2012). 3 Huang, W. Z. & Sloan, D. M. The pseudospectral method for solving differential eigenvalue problems. Journal of Computational Physics 111, (1994). 4 Weideman, J. A. C. & Reddy, S. C. A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, (2000).

17 (A) (B) (C) (D) (E) (F) Figure S1. Strips of biofilm-sediment composite were cut from the beds of the incubation flume and underwent load testing on a 5N Tinius O H1KS tensile Tester. (A) gives and example of a strip of biofilm-glass bead composite and (D) is a strip of biofilm-sand composite. A const displacement rate of 1.6 μm s -1 was applied and the load was recorded. The shape of the strip was digitized and imported into the COMSOL m physics finite element package where the constant rate of displacement was applied to one of the clamped boundaries and the load test simulated. (B) and (E) give examples of the simulated deformed shape of the composites at the end of a simulations. The Young s modulu

18 elasticity was calibrated so that the observed and simulated loads were in close agreement (C) and (F). This was done for three strips of biofilmglass bead composite and three biofilm-sand composite strips. The calibrated Young s moduli were Pa for beads and 7200 Pa for sand.

19 Table S1. Specification of the time resolved Dantec PIV system.

20 Q (l/s) U (m/s) Number of Images T (μs) Hz Time of Recording (s) Field of view (Pixels) Field of view (mm) x x x x x x x x Table S2. PIV set up for different flow rates.