Answer: Unfortunately, I don't think this will work. The cottonwood roots steal moisture and nutrients from the soil, the branches block rain from reaching the ground, and the tree also shades the area. Since the ornamental grasses need full sun and sufficient moisture to grow well, they are not good candidates for a location such as this.
When planting under a tree, in my experience the garden cloth can be problematic. The tree roots tend to become entangled in it and it prevents your spreading plants from spreading the way the naturally would. Its main benefit is in preventing weeds from rooting too far down, but in shade weeds are not usually that much of a problem. In general, using two inches of organic mulch should be ample weed prevention in a shady area. Try not to mulch over the larger surface roots.
Under a large tree, it is usually better to work with an undemanding groundcover that tolerates shade and relatively poor dry soil. They can be set in as small plants without disturbing so many tree roots and are good competitors in their own right. Vinca minor or periwinkle might be a good choice, or depending on how shady/sunny it really is, you might consider one of the spreading perennial geraniums such as Geranium sanguineum, or perhaps Ajuga. To give them a good start you will need to prepare each planting hole by loosening the soil and working in some organic matter and you will need to keep them weeded and watered until they become established. Using organic mulch between them until they fill in over the area will help the weeds down and will also help feed the soil gradually over time as it breaks down.
Your local professional nursery staff and/or county extension may have additional suggestions based on a more detailed understanding of the growing conditions around the tree and your design goals. Good luck with your project!
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