I think you have to be very careful about the Reynolds number occuring in the flow and the relative effects of viscous and inertial (turbulent) energy loss.  My suspicion is that the Re in the pumping left ventricle is >>1 in which case the viscous energy loss is relatively much smaller than the turbulent part. 

When I speak of turbulent energy loss Im speaking about the viscous losses associated with unstable flows which occur at scales which are smaller than those which see from your measurements (eddy cascades etc).

The consequence of this is that the integral of gammadot:Tau over the volume is dependent on the scale of the velocity field discretisation and you almost certainly vastly underestimate the energy losses.

Mr. ElBaz,

If I understand correctly, you have a 3D velocity field of the fluid. When calculating the rate of viscous dissipation, the velocity divergence (div.V) will be calculated using the signed vector representations of the velocity field.

For instance, if the velocity at the point (x, y, z) is given by (-vx)i + vyj + (-vz)k, the divergence will be computed as: div(V)x,y,z = (-dvx/dx) + (dvy/dy) + (-dvz/dz)

Please note that "d/dx" represents the partial derivative with respect to x (similarly so for y and z), as I am unable to type in the correct term "doh". Hope this helps.

Thanks Mr. Joshi for your answer.

Yes, I do understand the point that divergence has to be signed. Excuse me if I was not clear enough.  My question is about getting the total E-loss over all points in the domain of interest (In my case, Left ventricle ) i.e. when summing up the E-loss of all (x,y,z) mesh points should I sum the signed or unsigned E-loss? On other words, does negative E-loss represent something?

Thanks in advance

I apologize for the misinterpretation, Mr. ElBaz

The question can be answered by knowing whether the fluid in question is viscous, elastic or viscoelastic. Viscous fluids do not store energy, dissipating all of it in the form of heat. Therefore, for viscous fluids, we sum the unsigned E-loss.

Elastic/viscoelastic fluids, on the other hand, can store energy in themselves (hence the negative loss, i.e. gain of energy). Thus, for elastic/viscoelastic fluids, we sum the signed E-loss.

A quick read on blood's behavior revealed that it is viscoelastic in nature. Energy is stored in the deformation of RBCs. Therefore, for your problem, summing the signed E-loss is required.

Reference: http://en.wikipedia.org/wiki/Blood_Viscoelasticity#Blood_Viscoelasticity

Hello Mr. ElBaz,

you could look look up the mathematical formulation for dissipation from a standard textbook. Please find attached the dissipation rate for a newtonian fluid. By definition its local values cannot be negative. This is obvious by inspection of the formula which employs squared values for the respective components of the gradient field.

To compute the overall loss within a certain volume, a volume integration has to be performed. i.e   int(Phi)dV. This integral has to be approximated. You could therefore define discrete volume elements (presumably hexahedral) around your points at which the dissipation is computed, multiply the local values of the dissipation with the volume of the respective volume elements and sum up all products Phi(i)*Delta V(i): PowerLoss=sum_i(Phi(i)*Delta V(i)).

Best regards

Bastian

Dr. Schmandt,

In the image attached in your post, the second term on the right hand side of the equation contains three squared terms that each involve summation of two velocity gradients. It is the signs of these terms that need to be considered. If the fluid is viscous, we take the absolute value of the gradients. If elastic/viscoelastic, we consider their signs during summation.

Also, I believe a term is missing in the equation in the image: -(2*mu/3) * (del.V)^2

This term too has summation of the three kronecker velocity gradients.

Do I understand this correctly? I think stating "signed E-loss" was a bit ambiguous on my part. I have attached a reference document of detailed derivation. Please see on page 32 a section on energy dissipation and storage.

Hello Mr. Joshi,

i quoted the equation for incompressible flow. Actually two "+" signs are missing between the summands in the second brackets.

Why should one consider only the absolute values of the gradients irrespective of their sign? 

Best regards

Bastian

Thanks Mr. Schmandt for your answer.

I do agree with MR.Joshi that the divergence term might be missing from the equation in the image.

Thanks Mr. Joshi for your helpful and detailed answers. I am indeed aware of the reference you attached.

I also attached the reference which includes the equation I used in my question. please check Equation 2 therein which I use for computing non turbulent viscous energy loss. It might be important to mention that in our analysis we usually assume that blood is Newtonian, viscous and incompressible. 

Does incompressiblity assumption play a role in simplifying the dissipation function in the reference  pointed out by Mr. Joshi to  equation 2 in the attached reference?

Don't you think that under the incompressibilty assumption, the summation in equation 2  would be needed to be  over unsigned values?

Thanks

>Does incompressiblity assumption play a role in simplifying the dissipation function

In incompressible flow the divergence is zero.

Dear Dr. Schmandt and Mr. ElBaz,

I was wrong in my statement about the signs - it simply makes no sense to alter the signs of the terms themselves. Energy storage in viscoelastic materials (if and when it happens) requires a separate formulation (that incorporates oscillations).

To find the dissipation rate for Mr. ElBaz's problem, we need to sum 2*mu*(E-loss(i))^2 for all i.

Sorry for creating a confusion.

Dear Mr. Schamndt and Mr.Joshi,

Thank you for your help,discussion and clarifications as fluid mechanics is not my main specialty. The view is clearer now.

As a result, for incompressible flow, The energy dissipation =2 mu*S^2 where S is the strain rate tensor field and mu is the material viscosity.

Additionally, I would like to share with you the attached reference which seems to agree with the results of our discussion.

Thanks again and kind regards

Mohammed

I think you have to be very careful about the Reynolds number occuring in the flow and the relative effects of viscous and inertial (turbulent) energy loss.  My suspicion is that the Re in the pumping left ventricle is >>1 in which case the viscous energy loss is relatively much smaller than the turbulent part. 

When I speak of turbulent energy loss Im speaking about the viscous losses associated with unstable flows which occur at scales which are smaller than those which see from your measurements (eddy cascades etc).

The consequence of this is that the integral of gammadot:Tau over the volume is dependent on the scale of the velocity field discretisation and you almost certainly vastly underestimate the energy losses.

Thanks Dr. Burbidge for your answer.

Indeed, generally speaking, the flow data acquired with 4D Flow MRI are averaged over multiple cardiac cycles. That means, only the mean flow is acquired but not the turbulent flow. Also, the acquired resolution is usually low (about 4X4X4 mm3) and temporal resolution of around 30 ms. In my case, the flow velocities are further linearly (or cubic spline) interpolated  in space to get 1X1X1 mm3 instead. All this would indeed limit the possibility of computing the turbulent energy loss.

In our current work, our aim is to compare energy dissipation of cardiac left ventricle in  normal subjects to those of patients (with diastolic dysfunction). Therefore, as we only have  the average flow. We recognize that only nonturbulent viscous energy dissipation can be computed.

Our assumption is , if total energy dissipation (i.e. turbulent+ nonturbulent) was originally higher in patients compared to healthy subjects , then it might be that the nonturbulent  viscous energy (based on average flow) , although less pronounced, still sensitive or  able  to reflect this higher dissipation in those patients and to differentiate them from  healthy subjects? i.e. by comparing the nonturbulent viscous dissipation of patients and volunteers, difference would be still significant.

Do you think this could be a valid assumption? Any suggestions would be  appreciated.

Thanks in advance

Unsteady flow is not necessarily turbulent. Could you comment about the Reynolds number? If the flow was laminar and unsteady one could determine the instantaneous losses and compute the cycle average afterwards.

Best regards

Bastian

Thanks Dr. Schmandt for your answer.

Regarding the Reynolds number. First we need to consider that we are working with dynamic system (left ventricle ). Therefore, Reynolds number changes over the cardiac cycle.

In left ventricle, the peak Reynolds number usually happens at the peak diastolic filling and seems to range between 3000 and 10000 (Reynolds number computed as Re=U*D/v where U is the mean velocity of mitral valve and D is the mitral valve diameter at peak diastole and v is the dynamic viscosity of blood ). In other phases of the cycle (apart from peak diastole ) Re usually ranges between 1000 and 5000. 

By averaging instantaneous losses , do you mean nonturbulent viscous energy loss of every voxel averaged over the cardiac cycle?

Best regards,

Mohammed

Hello Mohammed,

for unsteady losses due to a bend i once did an averaging of the time dependent integral losses. For the Reynolds number you reported, however, i think Adam Burbridge is right and a part of your losses won't be resolved using piv.  Maybe a simulation can show the ratio of the unresolved part to the overall part? This could be used tentatively to derive an empirical correction factor for similar cases.

Best regards

Bastian